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🎯 Know Your Learning Goals
Welcome back! You already know the parts of a trapezoid. Today, you will investigate a special trapezoid and discover a useful relationship for finding lengths without measuring them directly.
By the end of this lesson, I can:
explain the base-angle and diagonal properties of an isosceles trapezoid;
describe and use the trapezoid midsegment theorem; and
find unknown angle measures and side lengths using trapezoid properties.
🧩 Explore and Build Understanding
Isosceles Trapezoid Properties
An isosceles trapezoid is a trapezoid whose legs are congruent. If ABCD is an isosceles trapezoid with AB ∥ CD and AD ≅ BC, then:
∠A ≅ ∠B and ∠C ≅ ∠D. Each pair of base angles is congruent.
AC ≅ BD. The diagonals are congruent.
Be careful: Angles on the same base are congruent. Angles along the same leg are generally supplementary because the bases are parallel so they are considered as same-side interior angles. Supplementary angles have a sum of 180°.
What is the Midsegment
A midpoint divides a segment into two congruent parts. Because X and Y are the midpoints of the legs, XY is the midsegment. The midsegment has two important properties:
It is parallel to both bases: XY ∥ EF and XY ∥ GH.
Its length is one-half the sum of the base lengths.
Midsegment = (Base 1 + Base 2)/2
If fractions are uncomfortable, use the equivalent form below. Both formulas communicate the same relationship.
2(Midsegment) = Base 1 + Base 2
See How It Works
Example 1: In isosceles trapezoid JKLM, ∠J and ∠M lie on the same base. If m∠J = 75°, find m∠M.
The base angles of an isosceles trapezoid are congruent, so they have equal measures.
m∠M = m∠J = 75°
Therefore, m∠M = 75°.
Example 2: A trapezoid has bases of 15 m and 25 m. Find the midsegment.
Step 1: Write the formula.
Midsegment = (Base 1 + Base 2)/2
Step 2: Substitute the base lengths.
Midsegment = (15 + 25)/2
Step 3: Add, then divide by 2.
Midsegment = 40/2 = 20 m
The midsegment is 20 m. This answer is reasonable because 20 lies between 15 and 25.
Example 3: The midsegment is 18 units and one base is 14 units. Find the other base.
Let x represent the unknown base. Use the fraction-free form.
2(18) = 14 + x
36 = 14 + x
22 = x
The other base is 22 units. Check: (14 + 22)/2 = 18.
Try it Yourself
In isosceles trapezoid ABCD, AB ∥ CD. If m∠A = 110°, find m∠B and explain the property used.
The bases of a trapezoid measure 12 cm and 24 cm. Find the midsegment.
A trapezoid has bases of 18 cm and 30 cm. Find the midsegment.
The midsegment is 16 cm and one base is 10 cm. Find the other base.
📒 Check Your Understanding
Quick Check
State two properties of an isosceles trapezoid.
Complete: The midsegment joins the __________ of the legs.
Find the midsegment when the bases are 16 cm and 28 cm.
The midsegment is 25 cm and one base is 18 cm. Find the other base.
Why should a midsegment length normally be between the two base lengths?
🚀 Choose Your Next Step
Do you need more support?
Use 2(Midsegment) = Base 1 + Base 2. First double the midsegment. Then subtract the known base to find the missing base. Always substitute the final answer into the original relationship to check it.
Do you want an additional challenge?
Create a real-life word problem involving an agricultural plot shaped like a trapezoid. Give two base lengths or one base and the midsegment. Include a labeled sketch, a complete solution, and a sentence explaining why the answer is reasonable.
🔓 Answer Keys
Show Answers to the Activity
Try it Yourself
110°
18 cm
24 cm
22 cm
Quick Check
Base angles are congruent; diagonals are congruent
midpoints
22 cm
32 cm
An average of two unequal numbers lies between them
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