3.5 Parallel Lines and Triangles Triangle Angle Sum Theorem: The sum of the measures of the angles of a triangle is 180 Triangle Exterior Angle Theorem: The measure of each Exterior angle of a triangle equals the sum of the measures of its opposite/remote interior angles.

Find x and y and z

Bell work

Geometry 4 examples

AB ≅ A ≅ BC ≅ B ≅ CD ≅ C ≅ DA ≅ D ≅

Step 3: What do you remember about the measures of a triangle? Example 2: Using Congruent Parts Suppose that WYS ≅ MKV. If the measure of angle W = 62 and the measure of angle Y = 35, what is the measure of angle V? Step 1: Draw the two triangles and label them with all information from the question. Step 2: Identify the congruent angles between the two triangles and label them with congruent marks. Step 3: What do you remember about the measures of a triangle? Step 4: Write an equation to find the missing angle.

Example 3

Theorem 4-1: Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. If: Then:

Proof of Theorem 4-1

Exit Ticket Ch4-1

Homeworks for Chapter 4

For Exercises 8, 9 & 12, can you conclude that the triangles are congruent? Justify your answer.

Using Algebra: Find the values of the variables.

Given Find the measures of the given angles of lengths of the given sides.

Find the Error!

Bell Work

4.2 & 4.3: Triangle Congruence by SSS, SAS, ASA, and AAS Side-Side-Side Postulate (SSS) If ___3 sides___ of one triangle are __congruent__ to ____3 sides____ of another triangle, then the two triangles are ___congruent__. If… Then…

Side-Angle-Side Postulate (SAS) If _2 sides_ and _the included angle_ of one triangle are _congruent_ to _2 sides_ and __the included angle__ of another triangle, then the two triangles are _congruent_. If… Then…

Example 1

Example 2

Day 3 (Nov 1, 2017)

Bell work

HW 4.2

HW 4.2 continued

HW 4.2 Cont.

Angle-Side-Angle Postulate (ASA) If _2 angles_ and the __included side__ of one triangle are _congruent_ to _2 angles_ and the __included side___ of another triangle, then the two triangles are _congruent_. If… Then…

Angle-Angle-Side Postulate (AAS) If _2 angles_ and a _ nonincluded side_ of one triangle are _congruent_ to _2 angles_ and the corresponding _ nonincluded side_ of another triangle, then the triangles are _congruent_. If… Then…

HW 4.3

HW 4.3

Day 4

BW #4, 10

Ch 4. 4. What does CPCTC stand for Ch 4.4 What does CPCTC stand for? “Corresponding Parts of Congruent Triangles are Congruent”

Example 1: Proving Parts of Triangles Congruent Draw the picture on the bottom right of page 244

Draw the picture on the bottom right of page 245

4.5 Isosceles and Equilateral Triangles PARTS OF THE ISOSCELES TRIANGLE __________________: congruent sides __________________: the third side __________________: formed by the two congruent legs __________________: the other two angles

Isosceles Triangle Theorem (4-3): If two __sides__ of a triangle are __congruent__, then the __triangles__ opposite those sides are congruent. If… Then…

Converse of the Isosceles Triangle Theorem (4-4): If two __angles_ of a triangle are __congruent_, then the _sides__ opposite those angles are congruent. If… Then…

Example 1: Using the Isosceles Triangle Theorems Draw the picture on the bottom right of page 251 Is angle WVS congruent to angle S? Explain. Is segment TR congruent to segment TS? Explain.

Theorem 4-5: If a line _bisects_ the vertex angle of an isosceles triangle, then the line is also the __perpendicular___ __bisector__ of the base.

Example 2: Using Algebra Draw the picture in the middle right of page 252. DO NOT LABEL ANGLE A 54! Suppose the measure of angle A = 27. What is the value of x?

If a triangle is equilateral, then the triangle is ___equiangular____. Corollary: a theorem that can be proved easily using another theorem Corollary to Theorem 4-3 If a triangle is equilateral, then the triangle is ___equiangular____.

If a triangle is equiangular, then the triangle is __equilateral__. Corollary to Theorem 4-4 If a triangle is equiangular, then the triangle is __equilateral__.