2 Answers

AGamePlayer · Answer 1 · 2018-04-29T23:55:50+0000

Answer: The missing step is,

m∠OCP ≅ m∠ABC because corresponding angles made by the same transversal on the parallel lines are congruent.

Explanation:

Given :
$AC \parallel BD$ ,
$AB \parallel CD$

And,
$AC \perp CD$

We have to Prove : Angle PCQ is complementary to angle ABC

⇒
$m\angle PCQ + m\angle ABC = 90^(\circ)$

Proof: Since,
$AC \perp CD$

⇒
$m\angle OCQ = 90^(\circ)$

But,
$m\angle OCQ = m\angle OCP +m\angle PCQ$ ( By angle addition postulate )

⇒
$m\angle OCP +m\angle PCQ = 90^(\circ)$ ( By transitive property of equality )

Since,
$AC \parallel BD$ ,

⇒
$\angle OCP\cong \angle ABC$ ( corresponding angles made by the same transversal are congruent)

⇒
$m\angle OCP = m\angle ABC$ ( By the definition of congruent angles )

This leads,

$m\angle ABC +m\angle PCQ = 90^(\circ)$ ( by the transitive property of equality )

Thus, by the definition of complementary angles,

Angle PCQ is complementary to angle ABC

Hence proved.

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