5.1 Congruent Triangles

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1 5.1 Congruent Triangles Two figures are congruent if they have the same and the same. Definition of Congruent Triangles ΔABC ΔDEF if and only if Corresponding Angles are congruent: Corresponding Sides are congruent: A B C AB BC CA 1. Write a congruence statement. 2. Given: XYZ RST. Name the 6 congruent corresponding parts. 3. Given: BLU MON. Find the value of x. 4. NPLM EFGH. Find the value of each variable. m L = 57, m M = 64, m U = (5x + 4) x = y = Geometry Chapter 4 Notes pg. 1

2 Third Angles Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent. 5. Solve for the value of x. Proving Triangles are Congruent CD, AB CD Given: AB E is the midpoint of BC andad. Prove: AEB DEC Geometry Chapter 4 Notes pg. 2

3 Proving Triangles are Congruent: SSS, SAS, ASA, AAS, and HL Warm Up DEF MNO. Complete the statements. 1. m E = m 2. DF = Use the given information to find the value of the variables. 3. ABC PQR 4. JKL XYZ Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. Geometry Chapter 4 Notes pg. 3

4 Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) Congruence Postulate If two angles and a NON-included side of one triangle are congruent to two angles and the corresponding NON-included side of a second triangle, then the two triangles are congruent. Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. Counterexample to show that Angle-Side-Side is not a valid reason to prove triangles are congruent. Geometry Chapter 4 Notes pg. 4

5 Identify which property will prove these triangles congruent. SSS SAS ASA AAS HL NONE WARM UP DAY 2 For each triangle, name the included angle for the sides given. 1. ABC: sides andac AB 2. DEF: sides DF anded State the congruence postulate or theorem that proves the triangles congruent. Then state the congruence statement T Geometry Chapter 4 Notes pg. 5

6 Proving Triangles are Congruent: SSS, SAS, ASA, AAS, and HL Day 2: Proofs BC, CD AB Prove: ADC BDC 1. Given: AC 2. Given: AD BC, AD BC Prove: ABC CDA Geometry Chapter 4 Notes pg. 6

7 3. Given: PQ bisects SPT SP PT Prove: SPQ TPQ 4. Given: AB AD, DE AD C is the midpoint of BE Prove: ABC DEC Geometry Chapter 4 Notes pg. 7

8 5. Given: AD CE, BD BC Prove: ABD EBC 6. Given: AB, CD AB CD Prove: ABC CDA Geometry Chapter 4 Notes pg. 8

9 5.5 Using CPCTC in Triangles How many triangles can you count in the diagram?? Five ways to prove triangles congruent: Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Given: ABC MNP 3. B 4. MP Once you have determined that two triangles are congruent, now you can say that all of the other corresponding parts are also congruent. 1. Given: AB, CD BC DA Prove: AB CD Geometry Chapter 4 Notes pg. 9

10 2. Given: A is the midpoint of MT A is the midpoint of SR Prove: MS TR 3. Given: Prove: BC DC Geometry Chapter 4 Notes pg. 10

11 4. Given: M N OKL OLK Prove: MK NL Geometry Chapter 4 Notes pg. 11

12 5.5 Day 2: Proofs with Isosceles Triangles & Equilateral Triangles Warm Up State which postulate or theorem you can use to prove that the triangles are congruent. Then explain how proving that the triangles are congruent proves the given statement. LMK NMK because LK NK because How many triangles are in the figure?? If an ISOSCELES triangle has exactly two congruent sides, then the congruent sides are the of the triangle and the noncongruent side is the. The two angles adjacent to the base are the. The angle opposite the base is the. Geometry Chapter 4 Notes pg. 12

13 Geometry Chapter 4 Notes pg. 13

14 Given: ABC is an isosceles triangle AB AC, AD bisects CAB Prove: B C Base Angles Theorem Two sides of a triangle are congruent if and only if the angles opposite them are congruent. A triangle is equilateral if and only if it is equiangular. 1. Find the value of x and y. 2. Find the values of x and y. Geometry Chapter 4 Notes pg. 14

15 Determine the values of x, y and z Solve for x and y. GUIDED PRACTICE Given: RV, ST RTV and SVT are right angles Prove: ΔRTV ΔSVT Geometry Chapter 4 Notes pg. 15

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