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2.8 – Triangle Theorems
Objectives
 Identify congruent parts of an isosceles triangle.
 Learn the isosceles triangle theorem and its converse.
 Discover the two corollaries associated with the isosceles triangle theorem that relate to equilateral triangles.
 Explore the relationship between the shortest and longest sides of a triangle to the smallest and largest angles of a triangle.
 Use triangle theorems to find missing side lengths and angle measures.
Key Terms
 Base – The side of the triangle that is perpendicular to the altitude
 Isosceles Triangle Theorem – A theorem stating that if 2 sides of a triangle are congruent, then the angles opposite those sides are congruent.
Notes
Theorems & Corollaries Chart 

Opposite Sides and Angles 
 Longest / Largest
 The longest side of a triangle is always opposite the angle with the greatest measure.
 The angle with the greatest measure is always opposite the longest side.
 Shortest / Smallest
 The shortest side of a triangle is always opposite the angle with the smallest measure.
 In a triangle, the angle with the smallest measure is always opposite the shortest side.
 Medium / Medium
 The medium angle will be opposite the medium side.

Isosceles Triangles 
 Have 2 or more congruent sides, called legs.
 Have 2 or more congruent angles, called base angles (opposite from the congruent sides).

Equilateral Triangles 
 Equilateral Triangles
 When all 3 sides of a triangle are congruent (equal lengths), then all 3 angles of the triangle will be congruent (equal measures).
 ALL equilateral triangles are isosceles.
 SOME isosceles triangles are equilateral.

 Example: Equilateral & Isosceles
 All 3 sides are 30 units in length
 Since all 3 sides are the same, at least 2 of them are the same (isosceles).
 All 3 angles are the same measure: 60 degrees (because 180 / 3 = 60)

Scalene Triangles 
 Scalene Triangles
 The sides lengths must all be different
 A triangle with two or more congruent angles can never be scalene
 The longest side of is
 The shortest side of is

Examples 
 Example 1: Missing Angle
 What is the measure of ∠ B, in degrees?
 Step 1: Notice that 2 sides are congruent in length, so this is isosceles
 Step 2: The opposite angles of those 2 sides will be congruent to one another
 Step 3: All 3 angles will add up to 180: (74 + 74 + B = 180)
 Step 4: Solve for ∠ B: (148 + B = 180 → Subtract 148 on both sides → B = 32°

 Example 2: Solve for x
 Step 1: Recognize that this is an isosceles triangle because both base angles are 63°
 Step 2: Recognize that the sides opposite base angles will be congruent (same length)
 Step 3: Since both sides are congruent, set up an equation where they are equal
 Set up the equation: 3x = x + 8
 Solve by subtracting 1x on both sides: 2x = 8
 Divide both sides by 2: x = 4
 Example 2b: What if you wanted to find the length of ?
 Step 1: Substitute x = 4 into either side
 Step 2: Solve: 3(4) = 12

 Example 3: Finding the shortest side
 In , , , . Which side of this triangle is the shortest?
 Step 1: Draw the triangle and label the verticies and angle measures
 Step 2: Notice which angle measure is largest
 Step 3: What side is opposite from that angle measure?
 Answer:

Proof 
Important!
Practice (Apex Study 2.8)
 Practice: Pgs 13, 14, 21, 22
 Mandatory: Draw and write the proof on Pg 6, 10
 Click through animations on Pgs 4, 5, 8
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