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4.2 Apply Congruence and Triangles

4.2 Apply Congruence and Triangles. You will identify congruent figures. Essential Question: What are congruent figures?. Tell students they will learn how to answer this question by studying the definition of congruent figures.

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4.2 Apply Congruence and Triangles

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  1. 4.2 Apply Congruence and Triangles • You will identify congruent figures. • Essential Question: What are congruent figures? Tell students they will learn how to answer this question by studying the definition of congruent figures

  2. Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. The diagram indicates that JKL TSR. Corresponding angles J T,∠ K S,L R Corresponding sides JK TS,KL SR,LJ RT EXAMPLE 1 Identify congruent parts SOLUTION

  3. In the diagram, DEFG SPQR. Find the value of x. Find the value of y. You know that FG QR. FG = QR = 2x – 4 12 16 = 2x 8 = x EXAMPLE 2 Use properties of congruent figures SOLUTION

  4. You know that∠ F Q. m F = mQ 68 = 6y + 8 68 o = (6y + x) o 10 = y EXAMPLE 2 Use properties of congruent figures

  5. PAINTING If you divide the wall into orange and blue sections along JK, will the sections of the wall be the same size and shape?Explain. From the diagram, A Cand D Bbecause all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC. EXAMPLE 3 Show that figures are congruent SOLUTION

  6. The diagram shows AJ CK, KDJB, and DA BC. By the Reflexive Property, JK KJ. All corresponding parts are congruent, so AJKD CKJB. EXAMPLE 3 Show that figures are congruent Then, 1 4 and 2 3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent.

  7. In the diagram at the right, ABGH CDEF. Identify all pairs of congruent corresponding parts. AB CD, BG DE, GH FE, HA FC A C, B D, G E, H F. for Examples 1, 2, and 3 GUIDED PRACTICE SOLUTION Corresponding sides: Corresponding angles:

  8. In the diagram at the right, ABGH CDEF. 2. Find the value ofxand findm H. (a) You know that H F (4x+ 5)° = 105° 4x = 100 x = 25 (b) You know that H F m H m F =105° for Examples 1, 2, and 3 GUIDED PRACTICE SOLUTION

  9. In the diagram at the right, ABGH CDEF. 3. Show thatPTS RTQ. SOLUTION All of the corresponding parts of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem. for Examples 1, 2, and 3 GUIDED PRACTICE

  10. FindmBDC. SOLUTION AB and ADCBCD, so by the Third Angles Theorem, ACDBDC. By the Triangle Sum Theorem, m ACD = 180°– 45° – 30°= 105°. So, mACD =mBDC = 105° by the definition of congruent angles. ANSWER EXAMPLE 4 Use the Third Angles Theorem

  11. Write a proof. GIVEN AD CB,DC AB CADACB ACDCAB, ACDCAB PROVE Use the Reflexive Property to show that AC AC. Use the Third Angles Theorem to show that BD EXAMPLE 5 Prove that triangles are congruent Plan for Proof

  12. STATEMENTS REASONS ,DC BA AD CB Given Reflexive Property of Congruence AC AC. ACDCAB, Given CADACB BD Third Angles Theorem ACDCAB Definition of EXAMPLE 5 Prove that triangles are congruent Plan in Action

  13. In the diagram, what is m DCN. CDN NSR, DNC SNRthen the third angles are also congruent NRS DCN = 75° for Examples 4 and 5 GUIDED PRACTICE SOLUTION

  14. By the definition of congruence, what additional information is needed to know that NDC NSR. ANSWER DC RS and DN SN for Examples 4 and 5 GUIDED PRACTICE

  15. In the diagram, ABCDEF. Complete the statement. 1. m A = ? ANSWER 60° 2. FD ? ANSWER CA 3. EDF ? ANSWER BAC Daily Homework Quiz

  16. Write a congruence statement for the two small triangles. Explain your reasoning. 4. s ANSWER WXZ YXZ; The diagram tell us that W Y and WZX YZX. WXZ YXZ by the Third Thm. From the diagram WX YX and WZYZ, and XZ XZ by Refl. Prop. Of Segs. Daily Homework Quiz

  17. • Triangles can be proved congruent by showing that all 3 pairs of corresponding sides and all 3 pairs of corresponding angles are congruent. • If two angles of one triangle are congruent to two angles of another, then the third angles are congruent. • You will identify congruent figures. • Essential Question: What are congruent figures? Congruent figures are figures that have exactly the same size and shape.

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