# 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 $\Delta DEF$ is $\overline{DF}$
• The shortest side of $\Delta DEF$ is $\overline{DE}$

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 $\overline{MN}$?
• Step 1:  Substitute x = 4 into either side
• Step 2: Solve:  3(4) = 12
• $\overline{MN}=12\quad units$

• Example 3: Finding the shortest side
• In $\Delta FGH$, $\angle F=65^\circ$, $\angle G=68^\circ$, $\angle H=52^\circ$. 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: $\overline{FG}$
Proof
 Prove ∠A ≅ ∠C 1. $\overline{AB} \cong \overline{BC}$ 1. Given 2. Draw D as the midpoint of $\overline{AC}$ 2. Every segment has a midpoint you can draw in 3. $\overline{AD} \cong \overline{CD}$ 3. Definition of a midpoint 4. Draw $\overline{BD}$ 4. You can connect any two points with a segment 5. $\overline{BD} \cong \overline{BD}$ 5.  Reflexive Property 6. $\Delta ABD \cong \Delta CBD$ 6. SSS Congruence Postulate 7. $\angle A \cong \angle C$ 7. CPCTC QED