Congruence between triangles
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This document defines and provides examples of congruent angles, congruent segments, and congruent triangles in geometry. It explains that angles are congruent if they have the same measure, segments are congruent if they have the same length, and triangles are congruent if corresponding angles and sides are congruent. The document provides exercises asking the reader to identify corresponding parts of congruent triangles without diagrams as well as included angles and sides within triangles.
Congruence between triangles
- 1. Which of these three-dimensional figures are congruent? Answer: (a) and (f); (b) and (d); (c)and (e)
- 2. Answer: (a) and (e); (b) and (c); (d)and (f)
- 4. DEFINITIONS Angles are CONGRUENT if they have the SAME MEASURE. ∠A ≅ ∠B means that m∠A = m∠B Ex. If m∠F = 34 and m∠G = 34, then ∠F ≅ ∠G Segments are CONGRUENT if they have the SAME LENGTH. AB ≅ CD means that AB = CD Ex. If EF = 2 cm and GH = 2 cm, then EF ≅ CD EQUATION is between NUMBERS or MEASUREMENTS. CONGRUENCE is between GEOMETRIC FIGURES.
- 5. We DON’T write “=“ between two names of geometric figures unless: D E C B A BA ∠BAC = ∠DAE AB = BA
- 6. B C D E FA ABC ⇿ DEF WHAT ARE THE CORRESPONDING SIDES AND CORRESPONDING ANGLES?
- 7. DEFINITION Given a correspondence between two triangles. If every pair of corresponding angles and corresponding sides are congruent, then the triangles are congruent.
- 8. EXERCISES 1. Given that ∆ABE ≅ ∆DCF, complete the following statements by supplying the missing symbols. The correspondence A___ ⇿ ___F is a congruence. B CA FE D ∠A ≅ ∠D ∠B ≅ ____ ∠E ≅ ____ AB ≅ ____ AE ≅ ____ BE ≅ ____
- 9. EXERCISES 2. Given that ∆MQP ≅ ∆NQP. List the six pairs of corresponding, congruent parts of these two triangles. M N P Q 3.For each of the congruences, list the six pairs of corresponding, congruent parts. a. ∆RQF ≅ ∆ABX. You may make a sketch of the triangles if you wish. b. ∆FHW ≅ ∆MRK. Do not use a figure. c. ∆DOG ≅ ∆CAT. Do not use a figure.
- 11. E BA F C D
- 13. E BA F C D
- 14. DEFINITION A side of a triangle is said to be INCLUDED by the angles whose vertices are the end points of the segment. An angle of a triangle is said to be INCLUDED by the sides of the triangle which lie in the sides of the angle.
- 15. A C B
- 16. EXERCISES 1. (a) In ∆ABC, which is the included angle of the sides BC and AB? (b) Which is the included side of ∠A and ∠C? (c) Which sides include ∠C? (d) Which angles include BC? 2. Consider ∆GHK. Without drawing a figure, can you discover an easy method of deciding which sides and angles are included sides and included angles? (a) Does GH and HK include ∠H? (b) Does ∠G and ∠K include GK? (c) What is the included angle of GH and GK? (d) What is the included side of ∠G and ∠H? A C B