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Zorluk: KolayProperties of Quadrilaterals

In isosceles trapezoid ABCDABCD, the parallel bases are ABAB and CDCD. If the measure of interior angle AA is 70∘70^\circ, what is the measure, in degrees, of interior angle CC?

Cevap: 110 degrees

Cevap

The measure of interior angle CC is 110110 degrees.
Since the trapezoid is isosceles with bases ABAB and CDCD, the base angles ∠A\angle A and ∠B\angle B are congruent, so ∠B=70∘\angle B = 70^\circ. The consecutive interior angles along the leg BCBC are supplementary because AB∥CDAB \parallel CD, which means ∠B+∠C=180∘\angle B + \angle C = 180^\circ. Solving for ∠C\angle C gives 180∘−70∘=110∘180^\circ - 70^\circ = 110^\circ.

Adım Adım Çözüm

1
Find the measure of angle BB using the properties of an isosceles trapezoid.
∠B=70∘\angle B = 70^\circ
In an isosceles trapezoid, the angles sharing a base are congruent. Since ABAB is a base, ∠A=∠B=70∘\angle A = \angle B = 70^\circ.
2
Calculate the measure of angle CC using the parallel lines property.
∠C=110∘\angle C = 110^\circ
Because the bases ABAB and CDCD are parallel, the consecutive interior angles ∠B\angle B and ∠C\angle C must sum to 180∘180^\circ. Therefore, ∠C=180∘−70∘=110∘\angle C = 180^\circ - 70^\circ = 110^\circ.

Anahtar Kavram

Properties of an isosceles trapezoid

Alternatif Yöntem

Since the sum of interior angles in any quadrilateral is 360∘360^\circ, and in an isosceles trapezoid the base angles are equal (∠A=∠B=70∘\angle A = \angle B = 70^\circ and ∠C=∠D\angle C = \angle D), we can write 70∘+70∘+∠C+∠D=360∘70^\circ + 70^\circ + \angle C + \angle D = 360^\circ. Since ∠C=∠D\angle C = \angle D, this simplifies to 140∘+2∠C=360∘  ⟹  2∠C=220∘  ⟹  ∠C=110∘140^\circ + 2\angle C = 360^\circ \implies 2\angle C = 220^\circ \implies \angle C = 110^\circ.
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Properties of Quadrilaterals

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