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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
GeoA 2.08 - Theorems
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.

GeoA 2.08 - Longest Side Largest Angle

GeoA 2.08 - Shortest Side Smallest Angle

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

GeoA 2.07 - Isosceles 03

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)

GeoA 2.07 - Equilateral01

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}

GeoA 2.08 - Scalene

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°

GeoA 2.07 - Isosceles 01

  • 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

GeoA 2.07 - Isosceles 02

  • 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

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