Isosceles Trapezoids & Midsegments
An isosceles trapezoid has congruent legs, giving it base angle congruence and congruent diagonals. The midsegment connects the midpoints of the legs.
Introduction
Isosceles trapezoid with midsegment shown in green
Past Knowledge
Trapezoid basics (9.3.1). Midsegments in triangles (6.1.1). Isosceles triangles (5.1.4).
Today's Goal
Apply properties of isosceles trapezoids and the midsegment formula.
Future Success
Kites (9.3.3), inscribed quadrilaterals, coordinate proofs (9.3.4).
Key Concepts
Isosceles Trapezoid Properties
- Legs are congruent (by definition)
- Base angles are congruent (both pairs: upper pair and lower pair)
- Diagonals are congruent
- Has a line of symmetry through the midpoints of the bases
Trapezoid Midsegment Theorem
The midsegment of a trapezoid:
- Is parallel to both bases
- Has length equal to the average of the bases
Theorem & Proof
Two-Column Proof: Base Angles of an Isosceles Trapezoid Are Congruent
Given: Isosceles trapezoid with ,
Prove: (lower base angles congruent)
| # | Statement | Reason |
|---|---|---|
| 1 | Drop perpendiculars from and to , creating feet and | Construction — perpendicular from point to line |
| 2 | (both equal the height ) | Perpendicular distance between parallel lines is constant |
| 3 | Given (isosceles trapezoid) | |
| 4 | HL Congruence (right triangles, steps 2, 3) | |
| 5 | CPCTC |
∎ Equal legs with equal heights create congruent right triangles, forcing the base angles to match.
Worked Examples
Midsegment Length
Trapezoid has bases 8 and 20. Find the midsegment.
Midsegment = 14
Finding a Base
Midsegment = 15, one base = 10. Find the other base.
→ →
Isosceles Trapezoid Angles
Isosceles trapezoid has ∠A = 75°. Find all four angles.
∠B = 75° (base angles congruent)
∠D = 180° − 75° = 105° (co-interior with ∠A)
∠C = 105° (base angles congruent)
∠A = ∠B = 75°, ∠C = ∠D = 105°
Common Pitfalls
Confusing Base Angles With Opposite Angles
In an isosceles trapezoid, the LOWER base angles are congruent to each other, and the UPPER base angles are congruent to each other. An upper and lower angle on the same leg are supplementary.
Thinking Midsegment Equals a Base
The midsegment is the average of the bases, not equal to either base. It always falls between the two base lengths.
Real-Life Applications
Stadium Seating
Each section of stadium seating is an isosceles trapezoid shape — wider at the back, narrower at the front. The midsegment formula helps calculate the number of seats in the middle row.
Lamp Shades
A typical lampshade is a truncated cone. Its cross-section is an isosceles trapezoid, and the symmetry ensures even light distribution.
Practice Quiz
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