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Triangle and Shape Correspondence: Vocabulary, Formulas, and Examples

Learn how to establish correspondence between triangles and shapes, determine similarity, and understand key vocabulary through formulas and examples.

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Triangle and Shape Correspondence: Vocabulary, Formulas, and Examples

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  1. Goal: I will be able to determine triangle/shape correspondence and similarity. Tool Bag Formulas, equations, Vocabulary, etc. Here’s How…Notes & Examples When studying triangles & shapes, it is essential to be able to communicate about the parts of a triangle/shape without any confusion. The following terms are used to identify particular angles or sides: Vocabulary • between • adjacent to • opposite to • the included (side/angle) Use figure △ABC to answer the following: B between 1. ∠A is ___________________ sides AB and AC. adjacent 2. ∠B is ___________________ side AB and side BC. opposite 3. Side AB is __________________ ∠C A C BC 4. Side _________ is the included side of ∠B and ∠C ∠B 5. ___________ is opposite to Side AC. ∠C ∠B 6. Side BC is between angles __________ and __________. ∠B 7. What is the included angle of AB and BC?

  2. Triangle Names Acute Triangle has 3 acute angles Acute Triangle Right Triangle has 1 right angle. Obtuse Triangle has 1 obtuse angle. 5” 5” Obtuse Triangle has 3 congruent sides. 5” Equilateral Triangle Right Triangle Equilateral Triangle has at least 2 congruent sides and 2 congruent angles. Isosceles Triangle 8” 8” 34° 34° Isosceles Triangle has no congruent sides and no congruent angles. Scalene Triangle Scalene Triangle

  3. Given two triangles, we need to determine a method to establish alignment between the sides and angles. B Correspondence The vertices of triangles/shapes are paired up. Y A C X We can chose to assign correspondence so that A matches to X, B matches to Y, and C matches to Z. Z There are 6 possible correspondences. Note: We use double arrows to show correspondence. A X A X A X B Y B Y B Y C Z C Z C Z A X A X A X B Y B Y B Y C Z C Z C Z We write the correspondence as follows: ABC XYZ ABC ABC XZY ABC YZX ABC ZXY ABC ZYX YXZ

  4. B Y A Why do we bother setting up correspondence? C A correspondence provides a systematic way to compare parts of two triangles. Without a correspondence, it would be difficult to discuss the parts of a triangle because we would have no way of referring to particular sides, angles, or vertices. X Z Assume the correspondence What can be concluded about the vertices? Vertex A corresponds to X, B corresponds to Y, and C corresponds to Z How is it possible for any two triangles to have a total of six correspondences? The first vertex can be matched with any of three vertices. Then, the second vertex can be matched with any of the remaining two vertices. ABC XYZ

  5. Example 1 Given the following triangles, use double arrows to show correspondence between vertices, angles, and sides. B T S A C R A S B T C R ∠A ∠S ∠B ∠T ∠C ∠R ABC STR AB ST BC TR CA RS

  6. Looking at the figure, it is impossible to tell the triangles apart. They look exactly the same. One triangle could be picked up and put on top of the other. A C B X Two triangles are identical if there is a triangle correspondence so that corresponding sides and angles of each triangle are equal in measurement. Z Y

  7. For triangles, it is useful to have a way to indicate those sides and angles that are equal. We mark sides with tick marks and angles with arcs if we want to draw attention to them. If two angles or two sides have the same number of marks, it means they are equal. F C B A E D In this figure… AC = EF = DE As you can see, tick marks are used to indicate equal sides and angles

  8. Example 2 Two identical triangles are shown. Give a triangle correspondence that matches equal sides and angles. T C B R A S Solution: You Try Draw two triangles that have correspondence. Have your partner check your work. ABC TSR

  9. Other Shapes Quadrilateral A four sided shape rectangle square trapezoid rhombus Pentagon A five sided shape Octagon A eight sided shape

  10. Example 3 Given the following quadrilaterals, find x, y, and z: PQSR WTUV T S 6y R 75° 132° W 18 WT PQ PR WV Find x: R V Find y: x + 5 x + 5 = 12 70° 12 75° 11z° 6y = 24 Q x + 5 - 5 = 12 - 5 P 24 V U x = 7 y = 4 Find z: 6y 24 132 = 11z = 6 6 12 = z 132 11z = 11 11

  11. Closing 1. Two shapes and their respective parts can be compared once a correspondence has been assigned to the two shapes. Once a correspondence is selected, corresponding sides and corresponding angles can also be determined. 2. Corresponding vertices are notated by double arrows; Shape correspondences can also be notated with double arrows. 3. Shapes are identical if there is a correspondence so that corresponding sides and angles are equal. 4. An equal number of tick marks on two different sides indicates the sides are equal in measurement. An equal number of arcs on two different angles indicates the angles are equal in measurement.

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