11
A property of the isosceles trapezoid
Textbook: pp. 32–33
GoalKnow that base angles of an isosceles trapezoid are equal and the segments cut off by an altitude.
New words
isosceles trapezoid · teng yonli trapetsiyalarger base · katta asossmaller base · kichik asosobtuse angle · o‘tmas burchak
Explanation
The base angles of an isosceles trapezoid are equal: ∠A = ∠D and ∠B = ∠C. In the proof we drop altitudes from B and C to the larger base; the two right triangles are congruent by hypotenuse and leg. Let the larger base be AD = a and the smaller base BC = b. The altitude from an obtuse-angle vertex cuts the larger base into segments of length (a − b) : 2 and (a + b) : 2. The diagonals of an isosceles trapezoid are equal too. If one angle is known, all the others follow at once.
Worked examples
The bases of an isosceles trapezoid are 10 cm and 24 cm. The altitude cuts the larger base into (24 − 10) : 2 = 7 cm and (24 + 10) : 2 = 17 cm.
An isosceles trapezoid has an obtuse angle of 115°. Base angles are equal, so its angles are 65°, 65°, 115°, 115°.
Class activity
“Symmetric trapezoid”: make an isosceles trapezoid from paper and fold it in the middle; notice that the two halves match.
Practice
1
The bases of an isosceles trapezoid are 8 cm and 20 cm. How long is the shorter segment the altitude cuts off the larger base (cm)?
6
2
In the same trapezoid, how long is the longer segment cut off by the altitude (cm)?
14
3
An isosceles trapezoid has an obtuse angle of 128°. How many degrees is an acute angle?
52
4
Why are the base angles of an isosceles trapezoid equal?
The two altitudes are equal and the legs are equal, so the right triangles are congruent and their acute angles are equal.