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2.8 – Triangle Theorems
- 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.
- 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.
|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.
- Have 2 or more congruent sides, called legs.
- Have 2 or more congruent angles, called base angles (opposite from the congruent sides).
- 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
- 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
- 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?
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