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Exploring Congruent Triangles: Solving Angles and Theorems in Geometry

This warm-up exercise focuses on identifying and calculating angles in congruent triangles using fundamental theorems. You'll learn to find unknown angles using the triangle sum theorem and the properties of congruence. The problem set includes finding measures of angles such as m∠ACB, m∠CLK, and m∠BCK, and identifying remote interior angles related to other angles. This activity reinforces understanding of congruent triangles, supplementary and remote interior angles, and key geometric theorems.

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Exploring Congruent Triangles: Solving Angles and Theorems in Geometry

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  1. Chapter 4Section 3 Exploring Congruent Triangles

  2. Warm-Up P K B • 1) Find m<ACB • 2) Find m<CLK • 3) Find m<CBP • 4) Which two angles are the remote interior angles for <MLK? • 5) Find m<BCK • 6) Find m<MLK 42 42 70 70 A C L M

  3. Warm-Up P K B • 1) Find m<ACB • All the angles in a triangle add up to 180. • 180 = 70 + 42 + m<ACB • 180 = 112 + m<ACB • 68 = m<ACB • 2) Find m<CLK • The third angle theorem states if two angles in a triangle are congruent to two angles in another triangle, then the thirds angles are also congruent.  • Since both triangles have angles measuring 42 and 70, m<ACB is congruent to m<CLK. • M<ACB = 68 = m<CLK 42 42 70 70 A C L M

  4. Warm-Up P K B • 3) Find m<CBP • <CBP and <ABC are supplementary. • 180 = m<CBP + m<ABC • 180 = m<CBP + 42 • 138 = m<CBP • 4) Which two angles are the remote interior angles for <MLK? • <LKC and <LCK 42 42 70 70 A C L M

  5. Warm-Up P K B • 5) Find m<BCK • <ACB, <BCK, and <KCL are supplementary. • 180 = m<ACB + m<BCK + m<KCL • 180 = 68 + m<BCK + 70 • 180 = 138 + m<BCK • 42 = m<BCK • 6) Find m<MLK • The exterior angle theorem states that the remote interior angles add up to the exterior angle. • m<MLK = m<LKC + m<LCK • m<MLK = 42 + 70 • m<MLK = 112 42 42 70 70 A C L M

  6. Vocabulary • Congruent Triangles-Triangles that are the same size and same shape. • CPCTC-Two Triangles are congruent if and only if their corresponding parts are congruent. • Congruence Transformations- If you slide, rotate, or flip a figure, congruence will not change. • Theorem 4-4- Congruence of triangles is reflexive, symmetric, and transitive. • * Use to indicate each correspondence.

  7. Example 1: If Triangle ABC is congruent to Triangle DEF, name the corresponding congruent angles and sides. B E A D C F AB DE BC EF AC DF <A <D <B <E <C <F

  8. Example 2: If Triangle RPQ is congruent to Triangle ZYX, name the corresponding congruent angles and sides. P X Y Q Z RP ZY PQ YX RQ ZX <R <Z <P <Y <Q <X R

  9. Example 3: If Triangle BIG is congruent to Triangle DEN, name the corresponding congruent angles and sides. Draw a figure and mark the corresponding parts. B BI DE IG EN BG DN <B <D <I <E <G <N I G D N E

  10. Example 4: If Triangle PQRis congruent to Triangle RST, name the corresponding congruent angles and sides. Draw a figure and mark the corresponding parts. T P PQ TS QR SR RR TR <R <T <Q <S <R <R Q S R

  11. Example 5: Complete the congruence statement. T Triangle TRV is congruent to _________. Triangle DEG B) Triangle XYZ is congruent to __________. Triangle PVZ C) Triangle YZW is congruent to __________. Triangle WXY R V D X P G E V Y X Y Z W Z

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