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CONGRUENT TRIANGLES Chapter 4 – Geometry B – Fall 2013 4.1 Triangles and Angles Classify triangles by their sides and angles. Find angle measures in triangles. Triangles Triangles are figures with three sides. We can classify triangles by their sides and angles. Classifying Triangles by Sides Classifying Triangles by Angles Example 1 Classify each triangle by its sides and angles. Angles in a Triangle The angles inside a triangle add up to 180°. Example 2: Find the value of x. Angles in a Triangle The measure of an exterior angle to a triangle is equal to the sum of the two angles that are not adjacent to it. Example 3: Find the value of x. Classwork Complete Worksheet 4.1 Practice B Homework 10/15 Complete Worksheet 4.1 Practice A Skip #1-5 4.2 Congruence & Triangles Identify congruent figures and corresponding parts. Write congruence statements. 4.1 Check Congruent Triangles Recall: Congruent means _________________ Congruent Triangles have 3 equal sides and 3 equal angles. When we write the congruence statement _______________ the order of the points matters! The order of the vertices should match (use the angles as a guide). Corresponding Sides & Angles Corresponding means “matching”. Corresponding Sides are the sides in two congruent triangles that are equal. Corresponding Angles are the angles in two congruent triangles that are equal. Identifying Corresponding Parts Identify the corresponding sides and angles in the two congruent triangles below. Example 1 Are the two triangles congruent? How do you know? Write a congruence statement and identify all corresponding sides and angles. Example 2 Is <C = <K? How do you know? Find the value of x. Homework 10/16 Complete Worksheet 4.2 Practice A Warm Up 10/17 Complete the problems on the worksheet independently! Classwork 10/17 Ch. 4.1: p. 198 #16-21, 31-39 Ch. 4.2: p. 206 #10-17, 22, 25 Homework: Study for Quiz! p. 210 #1-6 4.3 SSS & SAS Prove that triangles are congruent using SSS and SAS. Included Angle An included angle is the angle between two sides. Name the included angle between the pair of sides given: a. DF and FE b. DE and EF c. DF and ED Proving Triangles are Congruent We don’t always need to know that all the angles and all the sides are the same to say that two triangles are congruent. There are several shortcuts that we can use that only require us to know that 3 things are congruent. They work because knowing just some information can allow us to determine that all sides and angles are equal. SSS: Side-Side-Side If all three sides are congruent, then the triangles must be congruent. SAS: Side-Angle-Side If two sides and their INCLUDED ANGLE are congruent, then the triangles must be congruent. Examples Determine whether there is enough information to prove that the triangles are congruent. If yes, state the reason why you know they are congruent. Classwork 10/18 4.3 p. 216 #12-19 Homework 10/18 4.3 Worksheet Practice A 4.4 ASA & AAS Prove that triangles are congruent using ASA and AAS. ASA: Angle-Side-Angle If two angles and their INCLUDED SIDE are congruent, then the triangles must be congruent. AAS: Angle-Angle-Side If two angles and their NON-INCLUDED SIDE are congruent, then the triangles must be congruent. We CANNOT Use: AAA ASS Examples Determine whether there is enough information to prove that the triangles are congruent. If yes, state the reason why you know they are congruent. Examples Determine whether there is enough information to prove that the triangles are congruent. If yes, state the reason why you know they are congruent. Classwork 10/21 4.4 p. 223 #2-13 Homework 10/21 Triangle Congruence “Study Guide” – both sides! 4.5 Using Congruent Triangles Determine missing information needed to prove two triangles congruent. Digging Deeper… What information is missing that would tell me that two triangles are congruent? If I know that two triangles are congruent, what other sides and angles (not marked) do I now know are congruent? What information is missing that would tell me that two triangles are congruent? What information would I need to know to prove … By SAS? B By ASA? F E A D By AAS? C If I know that two triangles are congruent, what other sides and angles (not marked) do I now know are congruent? State the reason why the two triangles are congruent. Name all additional corresponding sides and angles. F B A C J E D G H Classwork Complete Worksheet 4.5 Practice A Homework Complete Congruent Triangles Worksheet 4.6 Properties of Special Triangles Use properties of isosceles, equilateral, and right triangles to find missing side and angle measures in triangles. Equilateral = Equiangular All equilateral triangles are equiangular. Or simply: All angles equal = All sides equal B A C Isosceles Triangles Sides Legs: The two equal sides Base: The third side Angles Base Angles: The two equal angles (on the base) Vertex Angle: Angle across from the base. Isosceles Triangle If we know two sides are equal, then the two base angles are equal (and vice versa!) Example 1: Solve for x and y. Isosceles Triangles Example 2: Solve for y. Right Triangles Legs: The two sides that make up the right angle. Hypotenuse: The side across from the right angle. Proving Right Triangles Congruent To prove that two right triangles are congruent, we only need to know that the hypotenuse and one of the legs are congruent. HL = Hypotenuse – Leg Examples Determine whether the following right triangles are congruent and state how you know. Classwork Complete Worksheet 4.6 Practice A Homework Complete Worksheet 4.6 Practice B