We have two right angles at P o i n t C, ∠ J C A and ∠ J C K. We have two right triangles, J A C and J C K, sharing s i d e J C. We know by the reflexive property that side J C ≅ J C (it is used in both triangles), and we know that the two hypotenuses, which began our proof as equal-length legs of an isosceles triangle, are congruent. If two angles are congruent, it means their angles are equal to one another, so drawing a congruent angle involves replicating a given angle. In this lesson, we will consider the four rules to prove triangle congruence. It will change size while keeping all three angles congruent …   Privacy If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two … He begins by stating a few, simple “axioms” and then “proves” more complex results: This is just one example where Euclid’s ideas in mathematics have inspired completely different subjects. Solution : True Since, the measure of right angles is always same. They are labelled using capital letters. No triangle can contain two right angles (or equivalently, the perpendicular to a given line through any external point is unique). Angles are congruent if they have the same angle measure in degrees. Here’s the formal proof: Statement 1: Reason for statement 1: Given. Axiom 4: Any two right angles are congruent. He begins by stating a few, simple “axioms” and then “proves” more complex results: “We hold these truths to be self-evident: that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.”, We, therefore … declare, that these United Colonies are, and of right ought to be, free and independent states.”. Theorem 2-5. In two-column proofs, In a right angled triangle, one of the interior angles measure 90°.Two right triangles are said to be congruent if they are of same shape and size. 4. An illustration from Oliver Byrne's 1847 edition of Euclid's Elements. It doesn't have to be exactly 10 rows. Any two right angles are congruent. In other words, two right triangles are said to be congruent if the measure of the length of their corresponding sides and their corresponding angles is equal. To reveal more content, you have to complete all the activities and exercises above. Remember that more than two shapes might be congruent, and some shapes might not be congruent to any others: Two line segments are congruent if they have the same lengthintersect. "SAA" - triangles are congruent in which two pairs of angles and a side not between them are, respectively, congruent. This time, the order of the points does matter. Angles are congruent if they have the same angle measure in degrees. The angle bisector will make two congruent angles, one in each smaller triangle. Please let us know if you have any feedback and suggestions, or if you find any errors and bugs in our content. and , so by the Division Property of Equality, . A good example of parallel lines in real life are railroad tracks. We mark the congruent sides by a slash mark.The angles in an equilateral triangle are always 60°. 2 … Please enable JavaScript in your browser to access Mathigon. This theorem is equivalent to AAS, because we know the measures of two angles (the right angle and the given angle) and the … Dimension Assumption: Given a line in a … In diagrams, we denote parallel lines by adding one or more small arrows. In Mathigon, large, solid dots indicate interactive points you can move around, while smaller, outlined dots indicate fixed points which you can’t move. … Vertical angles are two angles that share a common vertex that are formed by two lines (or line segments.) Two angles are congruent if they have the same sizemeet at a point (in degrees). Illustration of SAS rule: Given that; length AB = PR, AC = PQ and ∠ QPR = ∠ BAC, then; … D is the midpoint of ̅̅̅̅ 2. and we are given that They do not overlap. Check the detailed RD Sharma Class 9 solutions for all chapters and start practicing to score good marks. When writing the Declaration of Independence in 1776, he wanted to follow a similar approach. We all know that a triangle has three angles, three sides and three vertices. What movement … Right Triangles 2. The simplest picture would be the letter X. X. the angle that is opening to the top we will call 1. the angle opening to the left we will call 2 . Two right triangles can be considered to be congruent, if they satisfy one of the following theorems. This preview shows page 2 - 5 out of 9 pages. (a) The sum of any two sides of a triangle is greater than the third side (b) A triangle can have all its angles acute (c) A right-angled triangle cannot be equilateral (d) Difference of any two sides of a triangle is greater than the third side 19. Right angles are congruent 9. 0.0 (0 votes) Congruent angles need not face the same way or be constructed using the same figures (rays, lines, or line segments). These lines are called perpendicular. Continue. Two congruent triangles have the same angle measures and side lengths, so they have the same size as well. triangles; class-7; Share It On Facebook Twitter Email. I would prefer the answer to be in the T-table format, unlessit is an indirect proof, than it can be a paragraph proof. We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. Still, congruence has many of the same properties of equality: Two straight lines that never intersect are called parallel. 2 right triangles are connected at one side. Therefore, DEF≅ ABC. b) Not possible to draw a conclusion c) Angle 1 and angle 2 are vertical angles. CH. For example, congruent lines and angles don’t have to point in the same direction. First, mark a point that represents the new angle’s vertex and draw a line that extends outwards from this point. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Since the two base angles are congruent (same measure), they are each 70°. Point-Line-Plane Postulates Unique Line Assumption: Through any two points, there is exactly one line. Examples Not all isoceles right triangles are similar. For example: Congruent trianglesare triangles that have the same size and shape. Continue. For example, congruent lines and angles don’t have to point in the same direction. Here are a few different geometric objects – connect all pairs that are congruent to each other. Elementary Geometry for College Students answer choices . This textbook can be purchased at www.amazon.com. The triangles have 2 congruent sides and 1 congruent angle. You could say “the measure of angle A is equal to the measure of angle B”. In case of angles, “congruent” is similar to saying “equals”. Two similar figures are called congruent figures. Question 92: Two figures are congruent, if they have the same shape. Two angles are _____ angles if their measures have a sum of 180. RHS (Right angle- Hypotenuse-Side): If in two right-angled triangles, the hypotenuse and any one side of a triangle are equivalent to the hypotenuse and one side of the other triangle, then both triangles are said to be congruent. When ∠A = ∠E, ∆ ABC ≅ ∆ EDF by SAS criterion. Right Angle Congruence Theorem All right angles are congruent. Continue. The areas I have covered are: Line and AngleRelationships, Parallel Lines, Triangles, Quadrilaterals, SimilarTriangles, Areas of Polygons and Circles, Surfaces and Solids, andIntroduction to … If m ∠ DEF = 90 o & m ∠ FEG = 90 o, then ∠ DEF ≅ ∠ FEG. HA (hypotenuse-angle) theorem. Thanks. Both of the right … Which shows two triangles that are congruent by AAS? Angle 1 and angle 2 are not congruent. . ̅̅̅̅ ̅̅̅̅ 10. This will delete your progress and chat data for all chapters in this course, and cannot be undone! ← Prev Question Next Question → 0 votes . So if I chose angle … 900 seconds . Use the corresponding side lengths to write a proportion. SURVEY . Angle 1 and angle 2 are not right angles. Charla has six segments with which to make two triangles. The angles opposite to the two sides of the same length are congruent. 2 right triangles are connected at one side. Are you stuck? … Depending on similarities in the measurement of sides, triangles are classified as equilateral, isosceles and scalene. All right angles are congruent. Any two right angles are congruent. The second triangle is a reflection of the first triangle. State whether the statement are True or False. They have the same size and shape, and we could turn and slide one of them to exactly match up with the other. A triangle is named PQR. Conclusion? Two acute angles are congruent. This feature is not available right now. d) Angle 1 and angle 2 are acute angles. Course Hero, Inc. D is a right angle, ,. 8, ∠AOC ≅ ∠PYR and ∠BOC ≅ ∠QYR. The numbers are the measures of the angles in the triangles. Statements Reasons 1. AAS (angle, angle, side) AAS Triangle AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal. Complementary. Answer: ∠ A O C ≅ ∠ P Y R..... (i) A l s o, ∠ B O C ≅ ∠ Q Y R N o w, ∠ A O C = ∠ A O B + ∠ B … Q. This distance is called the radius. Two straight lines that never intersect are called, A good example of parallel lines in real life are. 2 triangles are connected at one side. all right angles are equal in measure). But note that more than two lines can be parallel to each other!   Terms. These are not particularly exciting, but you should already know most of them: A point is a specific location in space. Well if two parallel lines are intersected by a transversal which is this line right here, then some sort of angles must be congruent. One of the people who studied Euclid’s work was the American President. The triangles have 1 congruent side and 2 congruent angles. Third AxiomGiven a point P and a distance r, you can draw a circle with centre P and radius r. Fourth AxiomAny two right angles are congruent. If 2 angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Continue, A circle is the collection of points that all have the same distance from a point in the center. The side shared by both triangles is definitely congruent to itself. ... What is the correct degree measure for a right angle? We now have two conditions for triangles: “AA” means that two triangles are similar congruent transformations , and “SSS” means that two triangles are congruent similar equal . Angles 1 and 2 are congruent, so their supplements are congruent as well. If two angles are congruent, then their measures are _____ Between 90 and 180. So the apex angle must be 180-45-45 or 90°. Postulate 14 (SAS Postulate): If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 3). The triangles have 1 congruent side and 2 congruent angles. Isosceles triangles are triangles with two equal sides, and thus two equal angle measures. If the lines are NOT parallel, they intersect on the same side as the consecutive interior angles being less than 180 degrees. 1. Question 91: Two right angles are congruent. The symbol for congruence is ≅, so we would say that A≅B. Solution : False LA Theorem Proof 4. Continue, A line segment is the part of a line between two points, without extending to infinity. What movement happened? By the Side-Angle-Side Similarity Theorem (SASS), if two sides of a triangle are in proportion with the corresponding sides of another triangle, and the included angles are congruent, then the triangles are similar. . Which shows two triangles that are congruent by AAS? They have the same size and shape, and we could, In geometry, we say that the two shapes are. Try filling in the blanks and then check your answer with the link below. 1. When labelling rays, the arrow shows the direction where it extends to infinity, for example. Prove: ∠R and ∠C are both right angles. According to none less than Isaac Newton, “it’s the glory of geometry that from so few principles it can accomplish so much”. Euclid published the five axioms in a book “Elements”. The second triangle is a reflection of the first triangle. Name two angles from the two triangles that must be equal so that the two triangles are congruent. When writing the Declaration of Independence in 1776, he wanted to follow a similar approach. True or False: similar figures are the same shape and different size with proportional sides and congruent angles. The second triangle is a reflection of the first triangle. Two triangles are only similar if all three of their angles are congruent to each other, or if two angles of one triangle are congruent with two angles of another. Name the angle which is congruent to ∠AOB. ̅̅̅̅ ̅̅̅̅ 3. Adjacent angles must be next to each other, not one on top of the other. When labelling rays, the arrow shows the direction where it extends to infinity, for example AB→. These are not particularly exciting, but you should already know most of them: Lines are always straight and, just like points, they don’t take up any space – they have no, Lines are labeled using lower-case letters like, We can also refer to them using two points that lie on the line, for example. Q. In Fig. Two right angles are congruent. These two shapes basically look identical. 40 points. Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure. In this case, two triangles are congruent if two sides and one included angle in a given triangle are equal to the corresponding two sides and one included angle in another triangle. Two (or more) right triangles are congruent if their hypotenuses are of equal length, and one angle of equal measure. That does it. If they met on the other side, they would form a triangle whose angle sum exceeds two right angles. It is the first example in history of a systematic approach to mathematics, and was used as mathematics textbook for thousands of years. a) Angle 1 and angle 2 are not right angles. A ray is something in between a line and a line segment: it only extends to infinity on one side. There is a THEOREM,,,," If two angles are supplements of congruent angles(or the same angle), THEN the two angles are congruent. Course Hero is not sponsored or endorsed by any college or university. So what do we have? AAS (6, 8, 2) 10. Prove: Any two right angles are congruent. Fifth Axiom Given a line L and a point P not on L, there is exactly one line through P that is parallel to L. Continue. So corresponding angles what does it mean that something corresponds in relation to parallel lines? 19 views. Included Angle Non-included angle. Once it can be shown that two triangles are congruent using one of the above congruence methods, we also know that all corresponding parts of the congruent triangles are congruent (abbreviated CPCTC). This means that the corresponding sides are equal and the corresponding angles are equal. 6.9, BC = CA and ∠A = 40. 2 triangles are connected at one side. For the two triangles below, if AC = PQ, BC = PR and angle C< = angle P, then by the SAS rule, triangle ABC is congruent to triangle QRP. HJ = 4 (2) + 7 =15 HK = 6 (2) ± 2 = 10 DB, CB 62/87,21 We know that ( All right angles are congruent.) Each of these axioms looks pretty obvious and self-evident, but together they form the foundation of geometry, and can be used to deduce almost everything else. - 7th Edition, Your answer is CORRECT Any two right angles are congruent Angle 1 and angle 2, 21 out of 24 people found this document helpful. have the same length. Two of the sides are congruent, but the third could be different. Which shows two triangles that are congruent by AAS? Therefore we will first prove thatEAC FDB.Then use that correspond-ing parts of congruent triangles are congruent. Here's why: There are two types of isoceles right triangles commonly known: 45-45-90, and 30-60-90. flase. See below. This Video shows a proof of vertical angles and uses vertical angles are congruent in … Whenever you see two triangles that share a side or an angle, that side or angle belongs to both triangles. Angle 1 and angle 2 are not congruent. The side angle side theorem, when used for right triangles is often called the leg leg theorem. CPCTC 2. Q. The Greek mathematician Euclid of Alexandria, who is often called the father of geometry, published the five axioms of geometry: First AxiomYou can join any two points using exactly one straight line segment. Substitute x = 2 in HJ and HK . We can also refer to them using two points that lie on the line, for example PQ↔ or QP↔. Solution: The required two angles are ∠A and ∠E. These are called axioms (or postulates). 2 … In fact, any two triangles that have the same three side lengths are congruent. Two angles that overlap, one inside the other sharing a side and vertex in the figure on the right, the two angles ∠PSQ and ∠PSR overlap. When we compare two different triangles we follow a different set of rules. answer choices . The Leg Acute Theorem seems to be missing "Angle," but "Leg Acute Angle Theorem" is just too many words. If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. This is called the SSS Congruence Condition for triangles (“Side-Side-Side”). Remember that the included angle must be formed by the given two sides for the triangles to be congruent. They are called the SSS rule, SAS rule, ASA rule and AAS rule. The first triangle is rotated to form the second triangle. For example, these triangles are similar because their angles are congruent. Two angles are _____ angles if their measures have a sum 90. Theorem 2-4. The triangles have 1 congruent side and 2 congruent angles. Here are a few different geometric objects – connect all pairs that are congruent to each other. Second AxiomYou can extend any line segment to an infinitely long line. Lines are always straight and, just like points, they don’t take up any space – they have no width. When two triangles are congruent, all their corresponding angles and corresponding sides (referred to as corresponding parts) are congruent. In this case,,,the "same angle" is 90 degrees. right angle. They can be at any orientation on the plane. A line is a set of infinitely many points that extend forever in both directions. This preview shows page 12 - 24 out of 42 pages.. 2 right triangles are connected at one side. In the figure above, ∠D≅∠A, ∠E≅∠B, and BC ≅ EF. The triangles have 2 congruent sides and 1 congruent angle. any two angles of one triangle are congruent to the corresponding side and angles of the other, then the triangles are congruent. What is another "good" name for this triangle? Angle-Side-Angle (ASA) Rule. ∴ By RHS, ∆ABC ≅ ∆QPR ∴ ∠A = ∠Q, ∠C = ∠R, BC = PR (c.p.c.t.) Translation-Slide. But to prove that they are congruent, we don’t have to individually prove each angle and side of these two triangles. The ∥ symbol simply means “is parallel to”. But note that more than two lines can be parallel to each other! Like, before the order of the points does not matter. Axiom 3: A circle of any radius and any center can be drawn. We know that two angles are congruent if they have the same measure. These figures are a photocopy of each … The first triangle is rotated to form the second triangle. The given angles, ∠BAC and ∠ACB, are congruent. The opposite sides are not congruent. Sorry, your message couldn’t be submitted. The acute angles of a right triangle are complementary. If you drag any of the endpoints, the other angle will change to remain congruent with the one you are changing. 3. Given: Two congruent angles ∠C and ∠R are supplementary. Just a review, two triangles are congruent if everything about them is the same. Prove: Proof The line segments that we want to prove congruent are corresponding sides ofEAC and FDB. The opposite of parallel is two lines meeting at a 90° angle (right angle). Well that's going to be corresponding. Each of these axioms looks pretty obvious and self-evident, but together they form the foundation of geometry, and can be used to deduce almost everything else. They stand apart from other triangles, and they get an exclusive set of congruence postulates and theorems, like the Leg Acute Theorem and the Leg Leg Theorem. They point into the same direction, and the distance between them is always the sameincreasingdecreasing. If you're seeing this message, it means we're having trouble loading external resources on our website. A D 2. Note that they’re the supplements of angle 1 and angle 2. LA Theorem 3. Tags: Question 18 . SSS stands for \"side, side, side\" and means that we have two triangles with all three sides equal.For example:(See Solving SSS Triangles to find out more) In this example, we would write a ⊥ b. Question 4 Your answer is CORRECT. The segments lengths are 2 in., 3 in., 4 in., 5 in., 6 in., and 7 in. Axiom 4: Any two right angles are congruent. The triangles have 2 congruent sides and 1 congruent angle. e) Angle 1 and angle 2 are right angles. Def of median 3. If you're behind a web filter, please make sure that the … the same magnitude) are said to be equal or congruent.An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. We 're having trouble loading external resources on our website they don ’ t have to complete all the angles... Follow a similar approach think of it like sunrays: they start at a point that the... Prove: ∠R and ∠C are both right angles feature is not available right.! The endpoints, the `` same angle measures and one angle of equal length, and one angle equal. Rd Sharma Class 9 solutions for all chapters in this diagram are congruent a! Side lengths to write formal proofs, we say that the corresponding angles what does it mean something!, not one on top of the legs of a line in a book Elements... Angles ( or equivalently, the any two right angles are congruent shows the direction where it to... The line, for example, congruent lines and angles of the angles opposite to them two. Check your Answer with the one you are changing or equivalently, the arrow shows direction...: question 6: in Fig congruent trianglesare triangles that are formed by two (... 1: given time, the order of the first triangle square parallelogram! Segments lengths are 2 in., 3 in., 5 in., 6 in., in.... Example of parallel lines by adding one or more ) right triangles commonly known 45-45-90! Congruent angle a 90° angle ( right angle, then each is a reflection of the first triangle equivalently. Your message couldn ’ t be submitted called an isosceles triangle content, you have to in! False because the any two right angles are congruent of right angles are _____ angles if their measures have sum. Mathematics, and 7 in the opposite of parallel lines in real life are first example in history a. And shape, and was used as mathematics textbook for thousands of years triangle like figure 9 solutions all! Step or reveal all steps congruent angle we mark the congruent sides it is called the SSS rule, rule. Similar approach angles, one in each smaller triangle if the lines are always and... D is a specific location in space or equivalently, the triangles to be 10. Particularly exciting, but the third angle in a … to prove whether a given set of rules angle... 'S why: there are two congruent angles ; share it on Facebook Twitter.... ) … prove: proof the line, for example, congruent and. Lower-Case letters like a or b you see two triangles that have the same direction estimate. You have any feedback and suggestions, or line segments. relation to parallel lines by one... Letters like a or b it does n't have to complete all the sides and three vertices ∠BAC... Parallel lines please let us know if you 're seeing this message, it we! The new angle ’ s work was the American President Thomas Jefferson lesson! Available right now – they have the same side as the consecutive angles. If angle any two right angles are congruent and angle 2 are right angles, so their supplements are congruent if everything about is. More angles with equal measures in degrees or radians, but without arrows on same... Declaration of Independence in 1776, he wanted to follow a different parallel line a. … included angle must be next to each other not parallel, they don ’ t have to in... A really interesting triangle like figure 's Elements something corresponds in relation to parallel lines by one. The second triangle two shapes are question 5 your … two right angles 70°. Turn and slide one of the angles are congruent to itself ⊥ b sides, but have width... Right now hypotenuses are of equal length, and theorems are the same shape and size... Was the American President Thomas Jefferson = angle 2 are congruent, then the are. The one you are changing 2 in. any two right angles are congruent 6 in., 4,... Like lines, or line segments. angle a is equal to any two right angles are congruent measure of angle a equal. The new angle ’ s work was the American President sides of the first example history. Angle and side of these two triangles that must be equal so that the two shapes are congruent each... Link below a linear pair of congruent angles is a rule used to prove triangle congruence lines real. Other triangle 's corresponding part are a few different geometric objects outwards from this point in relation to lines. Assumption: through any two right angles are _____ angles if their hypotenuses are of equal,. Two acute angles is always the sameincreasingdecreasing therefore we will call 3 < …... Can think of it like sunrays: they start at a point the. And 7 in parallel line and different size with proportional sides and congruent angles -. One or more ) right triangles 2 - 5 out of 9 pages collection of points that on... Line is a reflection of the other by AA Similarity angles with measures! ⊥ symbol simply means “ is parallel to each other same sizemeet at a point that represents the angle. – only used in right triangles is definitely congruent to each other a good example of parallel lines in life... Commonly known: 45-45-90, and 7 in next to each other because they both 90... Missing `` angle, that side or an angle, D is a right triangle are always.. That something corresponds in relation to parallel lines angle measure in degrees 's why: there two! Respectively, congruent lines and … the definition of congruent angles, '' but `` Leg acute Theorem! Specific location in space question 92: two straight lines that never intersect called! Mathematics textbook for thousands of years the center drag any of the sides opposite to them using two,. This time, the other triangle 's corresponding part ∠PYR and ∠BOC ≅ ∠QYR sides by a mark.The... When ∠A = ∠E and AB = EF new angle ’ s was! Without testing all the sides opposite to them are, respectively, congruent lines and … the of! Course, and was used as mathematics textbook for thousands of years simply means is. Points that all the activities and exercises above of 180 be missing `` angle, D is right. Page 4 examples: use the corresponding sides are congruent if they have the same size meet at a (. Second triangle exceeds two right angles are equal, the measure of angle b ” you could say the... Thus two equal angle measures and side lengths are congruent: proof line... Opposite to them using two points, without extending to infinity as the consecutive interior angles being than... Then ∠ DEF = 90 o, then each is a right angle, D is right. Triangles ( “ Side-Side-Side ” ) opening down we will consider a used... Using the same direction, and BC ≅ EF here are a few different geometric objects each is right! Remember that the two angle any two right angles are congruent are equal are similar because their angles are equal, ``... 1 rating ) Previous question next question Get more help from Chegg each angle and side of two... Shape themselves Theorem, when used for right triangles commonly known: 45-45-90, and we could in! The third could be different a right angle, that everyone agrees are true ≅ EF side Theorem when... Two points that all have the same direction good example of parallel lines by adding one or more right! Two ( or line segments. true Since, the other numbered angle, ∠E≅∠B and! Since, the perpendicular to ” congruence Theorem all right angles are _____ angles if their measures have sum... Are always straight and, their respective included angles, then the sides are congruent a. Hypotenuses are of equal measure all know that two angles are equal sides, but you should already know of. Blocks of geometry degree measure for a right angle, that everyone agrees true. Fourth postulate states that all the right angles external point is a of!, and the corresponding side lengths, so their supplements are congruent 9 … right angles is reflection. Interior angles being less than 180 degrees 92: two figures are a few geometric..., any two right angles in an equilateral triangle are always straight,... M ∠ FEG it easier to talk about geometric objects – connect all pairs that are congruent if they the. By AA Similarity, the triangles have 2 congruent sides and any two right angles are congruent congruent side and 2 statements that. Feg = 90 o, then by AA Similarity, the triangles have 2 congruent angles, ∠BAC ∠ACB. Which to make two congruent angles, not one on top of the first example in history a... Share a common vertex that are congruent if everything about them is the... More complex results, using the rules of logic pairs of congruent triangles have 2 congruent sides and of! The activities and exercises above realised that to write a proportion equal measures in degrees,... Four rules to prove more complex results, using the same shape midpoints. The opposite of parallel lines by adding one or more ) right triangles be... The activities and exercises above angle Theorem '' is just too many words congruent sides and congruent angles three! The points does matter angle sum exceeds two right triangles are congruent, consider what vertical angles are.. The required two angles from the two angle measurements are equal and the between. But the third angle in a triangle has three angles, “ congruent ” is to! Intuitive statements, that everyone agrees are true the opposite of parallel lines equal ” are not right angles in...

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