Chapter 4 Triangles Overview

Transcription

1 Chapter 4 Triangles Overview Ohio State Standards for Mathematics: G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA,SAS,and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.10 Prove theorems about triangles. (Including measures of interior angles of a triangle sum to 180, and base angles of isosceles triangles are congruent) G.GPE.4 Use coordinates to prove simple geometric theorems. Specific Learning Targets by Book Section (I can ) 4.1& 4.6 Triangles and Angles Classify triangles by their angles and sides. (ex. Acute scalene) Name parts of a triangle. (ex. Vertex, legs, base, hypotenuse) Find missing interior and exterior angles of a triangle. Use the properties of congruent sides and angles to find missing side lengths or angles in special triangles. 5.5 & 5.6 Triangle Inequalities Use triangle measurements to decide which sides and angles are largest and smallest. Compare sides and angles of two triangles using the Hinge Theorem. 4.2 Congruence and Triangles Identify corresponding parts of two congruent triangles and use this information to solve for missing sides and angles in a triangle. Prove that 3 sides and 3 angles are congruent in two triangles to prove that the two triangles are congruent triangles. 4.3 Proving Triangles Congruent: SSS and SAS Prove two triangles congruent by only showing the 3 corresponding sides are congruent (SSS). And write this in a formal proof. Prove two triangles are congruent by only showing 2 corresponding sides and their included angle are congruent (SAS). And write this in a formal proof. 4.4 Proving Triangles Congruent: ASA and AAS Prove two triangles congruent by only showing 2 corresponding angles and their included side are congruent (ASA). And write this in a formal proof. Prove two triangles are congruent by only showing 2 corresponding angles and a non-included side are congruent (AAS). And write this in a formal proof. 4.5 Using Congruent Triangles *** After proving that two triangles are congruent (Lessons ), then prove that any other missing angles or sides are also congruent according to the definition of congruent triangles. 4.6 Isosceles, Equilateral, and Right Triangles Prove that two right triangles are congruent by showing one leg and the hypotenuse are congruent (HL congruence theorem). And write this in a formal proof. 4.7 Triangles and the Coordinate Plane Use proper placement of geometric figures in the coordinate plane to make finding angles and distances of segments easy. (utilize the origin and axes) Write a coordinate proof by using distance formula (or Pythgorean Theorem) to find segment lengths and prove that two sides of a triangle are congruent.

2 4.1/4.6 Classifying Triangles Learning Targets: I can Classify triangles by angles and sides and use correct triangle vocabulary Vocabulary Word Acute Triangle Diagram Equiangular Triangle Right Triangle Obtuse Triangle Equilateral Triangle Isosceles Triangle Scalene Triangle

3 Special Triangles: Isosceles Triangle Right Triangle Equiangular Equilateral Triangle

4 Example 1: Classify the triangle by angle and side. Be as specific as possible. Example 2: Identify an isosceles triangle and an equilateral triangle. Example 3: Example 4: Triangle RST is equilateral. Find the value of x, and the length of each side. Example 5: Triangle CED is shown below. Classify the triangle by angles and sides.

5 4.1/4.6 Finding Angle Measures in Triangles Learning Targets: I can Find interior/exterior angle measures in triangles Exterior Angles Theorem An exterior angle of a triangle is equal to the sum of the two non-adjacent (remote) interior angles of the triangle. Example 1: Find the missing angle measures.

6 Example 2: Find the missing angle measures. Example 3: The angles of triangle are shown below. Find the measure of each angle. Then classify the triangle by its angles. Example 4: Find the missing angle measures below. Example 5: Find the missing angle measures below.

7 5.5/5.6 Triangle Inequalities Learning Targets: I can Use triangle measurements to decide which side is longest or which angle is largest Use the triangle inequality theorem to determine if three sides will form a triangle Use the Hinge Theorem to compare side lengths and angle measures. Triangle Inequality Theorem:

8 Example 1:

9 Hinge Theorem: Converse of the Hinge Theorem: Examples:

10 4.2 Congruence and Triangles Learning Targets: I can Identify congruent figures and corresponding parts Prove that two triangles are congruent Example 1: Example 2: Example 3: Find the value of x. Third Angles Theorem

11 Definition of Congruent Triangles: Example 4: Determine if the triangles below are congruent. Justify your reasoning. Example 5: Are the following triangles congruent?

12 4.3/4.4 Proving Triangles Congruent Learning Targets: I can Prove the two triangles are congruent using SSS, SAS, ASA, or AAS. Side-Side-Side Congruence Postulate (SSS) Side-Angle-Side Congruence Postulate (SAS) Angle-Side-Angle Congruence Postulate (ASA) Is there a SSA Congruence Postulate?? Angle-Angle-Side Congruence Theorem (AAS or SAA)

13 Informal Proofs: Decide whether enough information is given to prove the triangles congruent. a. b. c. Formal Proofs: Use a two-column, paragraph, or flow proofs to prove the triangles congruent. a. What is your plan? What postulate can you use? b. Given: WY XZ, Y X Prove: WYZ ZXW c. Given: QRP SRT; Q and S are right angles; R is the midpoint of QS Prove: QRP SRT

14 4.5 Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Learning Targets: I can Prove that parts of congruent triangles are congruent. Example 1: Given: WYX ZXY, WXY ZYX Prove: WX ZY Example 2: Given: SD TC, CS DT Prove: SCT TDS Example 3: In the figure shown, LN and MO bisect each other at point P. Fill in the proof to show that M O.

15 4.6 Using SSA to prove congruent triangles Learning Targets: I can Prove that two triangles are congruent using HL-Congruence Theorem Prove statements about triangles on the coordinate plane I. Proof using HL Theorem Example 1: Example 2:

16 4.7 Proof on the Coordinate Plane Example 3: Example 4: If Z is a midpoint of WY, prove WXZ YXZ. Example 5: Find the missing coordinates use only the letters provided Example 6: