Question

given:
ad = bc
bc ⊥ ae
ad ⊥ be
prove:
∠a = ∠b

1. ad = bc, bc ⊥ ae, ad ⊥ be
2. ∠d and ∠c are right angles
3. ∠e = ∠e
4. triangle ade congruent to triangle bce
5. ∠a = ∠b

Explanation:

Step1: Identify the reason for step 1

The first step lists the given information. So the reason is "Given".

Step2: Identify the reason for step 2

Since \(BC\perp AE\) and \(AD\perp BE\), by the definition of perpendicular lines (perpendicular lines form right angles), \(\angle D\) and \(\angle C\) are right angles.

Step3: Identify the reason for step 3

\(\angle E=\angle E\) is due to the reflexive property (a quantity is equal to itself).

Step4: Identify the reason for step 4

We have \(AD = BC\) (given), \(\angle D=\angle C\) (right angles from step 2), \(\angle E=\angle E\) (reflexive from step 3). So by the LA (Leg - Angle) congruence criterion for right - angled triangles (in right - angled triangles, if a leg and an acute angle of one right - angled triangle are equal to the corresponding leg and acute angle of another right - angled triangle, the triangles are congruent), \(\triangle ADE\cong\triangle BCE\).

Step5: Identify the reason for step 5

Since \(\triangle ADE\cong\triangle BCE\), by CPCTE (Corresponding Parts of Congruent Triangles are Equal), \(\angle A=\angle B\).

Answer:

1. Given
2. Perpendicular lines form right angles.
3. Reflexive
4. LA
5. CPCTE