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Geometry—Mrs. Dubler Chapter Four—Congruent Triangles Section 4.1 Triangle Triangles and Angles Classification of Triangles By Sides 2. 1. 3. Classification of Triangles by Angles 1. 3. 4. 2. Example 1: When classifying triangles you need to be as specific as possible. Vertex Adjacent sides Legs a. ΔABC has three acute angles and no congruent sides. Classify the triangle. b. ΔDEF has one obtuse angle and two congruent sides. Classify the triangle. hypotenuse base Example 2: Draw an isosceles obtuse triangle. Draw a scalene acute triangle. Example 3: Classify ΔPOQ by its sides, then determine if it is a right triangle or not. Interior angles Exterior Angles Triangle Sum Theorem Exterior Angle Theorem The __________ of the measures of the ____________________ angles of a triangle is ______________. The measure of an ________________ angle of a triangle is ____________ to the ___________ of the measures of the two _________________________ _________________ angles. Example 4: Find the value of x. Corollary Corollary to the Triangle Sum Theorem Example 5: The ____________ angles of a ______________ triangle are ______________________. Find the value of x. Section 4.2 Congruent figures Corresponding angles and sides Example 1: ΔABC≅ΔFED. List all congruent sides and angles. Congruence and Triangles Example 2: In the diagram DEFG≅SPQR. a. Find the value of x. b. Find the value of y. Third Angles Theorem Example 3: Find m<BDC. Example 4: Prove triangle ABC≅CDA If two ______________ of one triangle are ______________ to two ____________ of another triangle, then the ___________ angles are also ____________________. Example 5: Decide whether or not the triangles are congruent. Justify your reasoning. Properties of Congruent Triangles Reflexive Property Symmetric Property Section 4.3 Proving Triangles Congruent SSS and SAS If __________________ sides of one triangle are __________________ to ________________ sides of a second triangle, then the two triangles are ____________________. Side-Side-Side Congruent Postulate Example 1: Prove that ΔLMK≅ΔLMN Transitive Property Example 2: Which are the coordinates of the vertices of a triangle congruent to ΔPQR? a. b. c. d. (-1,1)(-1,4)(-4,5) (-2,4)(-7,4)(-4,6) (-3,2)(-1,3)(-3,1) (-7,7)(-7,9)(-3,7) Side-Angle-Side Congruence Postulate If two _____________ and the _________________ angle of one triangle are _______________ to two _____________ and the __________________ angle of a second triangle, then the two triangles are ___________________. Example 3: Prove that ΔABC≅ΔCDA. Example 4: In the diagram QS and RP pass through the center of the circle. What can you conclude about ΔMRS and ΔMPQ? Section 4.4 Angle-Side-Angle Congruence Postulate Proving Triangles are Congruent ASA and AAS If two _________________ and the ________________ side of one triangle are __________________ to two _______________ and the _________________ side of second triangle, then the two triangles are ______________________. Example 1: Example 1: Tell if the triangles can be proven congruent. Angle-Angle-Side If two _____________ and a _____________________ side of one triangle are _________________ to two angles and the ______________________ ____________________ side of another triangle, then the two triangles are __________________. Example 2: Prove ΔBAC≅ΔEDF without using AAS. Hypotenuse Leg Congruence Theorem If the ____________________ and a ________ of a __________ triangle are _________________ to the _________________ and _________ of a second right triangle, then the two triangles are ______________________. Example 3: ̅̅̅̅̅ ≅ ̅̅̅̅ Given:𝑊𝑌 𝑋𝑍, ̅̅̅̅̅ ̅̅̅̅ 𝑊𝑍 ⊥ 𝑍𝑌, 𝑎𝑛𝑑 ̅̅̅̅ ⊥ 𝑍𝑌. ̅̅̅̅ 𝑋𝑌 Prove: ΔWYZ≅ΔXZY Example 4: Are the pairs of triangles congruent? If so, justify your answer. Example 5: In the diagram, ̅̅̅̅ 𝐶𝐸 ⊥ ̅̅̅̅ 𝐵𝐷 ∠𝐶𝐴𝐵 ≅ ∠𝐶𝐴𝐷 Show ΔABE≅ΔABD Section 4.6 Base angles of an isosceles triangle Vertex Angle of an isosceles triangle Base Angles Theorem Converse of the Base Angles Theorem Example 1: In ΔDEF, ̅̅̅̅ 𝐹𝐷 ≅ ̅̅̅̅ 𝐸𝐷 Name two congruent angles. Isosceles, Equilateral, and Right Triangles If two ________ of a triangles are ______________, then the ___________ ______________ them are ________________. If two ______________ of a triangle are ____________, then the _______________ _____________ them are _______________ Example 2: a. If ̅̅̅̅ 𝐻𝐺 ≅ ̅̅̅̅,then 𝐻𝐾 _____≅_____ b. If ∠𝐾𝐻𝐽 ≅ ∠𝐾𝐽𝐻, then _____≅_____. Corollary to the base angles theorem Converse corollary to the base angles theorem If a triangle is _____________, then it is _______________ If a triangle is ____________, then it is __________________. Example 3: Find the measures of ∠𝑃, ∠𝑄, 𝑎𝑛𝑑 ∠𝑅. Example 4: Find the values of x and y in the diagram. Example 5: In the lifeguard ̅̅̅̅, tower ̅̅̅̅ 𝑃𝑆 ≅ 𝑄𝑅 𝑎𝑛𝑑 ∠𝑄𝑃𝑆 ≅ ∠𝑃𝑄𝑅. a. What congruence postulate can you use to prove ΔQPS≅ΔPQ R? b. Explain why ΔPQT is isosceles. c. Show ΔPTS≅ΔQT R. Section 4.7 Triangles and Coordinate Proof Coordinate Proof Example 1: Place a 2 unit by 6 unit rectangle in a coordinate plane. Example 2: A right triangle has legs of 5 units and 12 units. Place the triangle in the coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse. Example 3: Place an isosceles triangle in a coordinate plane. Find the length of the hypotenuse, and the midpoint, M. Example 4: Right ΔOBC has leg lengths of h units and k units. You can find the coordinates of points B and C by considering how the triangle is placed in the coordinate plane. Find the length of the hypotenuse ̅̅̅̅ . 𝑂𝐶 C (h,k) h units O k units B (h,0)