Question

ABCD is a rhombus in which altitude from vertex D to side AB bisects AB. The measures of angles of the rhombus are ________.

Answer 1

Given:
ABCD is a rhombus
Altitude from vertex D to side AB bisects AB.

Let DE biescts AB.
⇒ AE = EB ...(1)

In ∆ADE and ∆BDE

DE = DE (Common side)
∠AED = ∠BED (Right angle)
AE = EB (from (1))

By SAS property, ∆ADE ≅ ∆BDE

Thus, AD = BD (By C.P.C.T) ...(2)

Since, sides of a rhombus are equal
Thus, AD = AB ...(3)

From (2) and (3)
AD = BD = AB
Therefore, ∆ADB is an equilateral triangle.

Hence, ∠A = 60°
∠A = ∠C = 60° (opposite angles of rhombus are equal)

Sum of adjacent angles of a rhombus is 180°
⇒ ∠A + ∠D = 180° and ∠C + ∠B = 180°
⇒ 60° + ∠D = 180° and 60° + ∠B = 180°
⇒ ∠D = 180° − 60° and ∠B = 180° − 60°
⇒ ∠D = 120° and ∠B = 120°

Hence, the measures of angles of the rhombus are 60°, 120°, 60° and 120°.