ABCD is a rhombus in which altitude from D to side AB bisects AB. Find the angles of the rhombus.
Solution:
Given, ABCD is a rhombus
The altitude from D to side AB bisects AB
We have to find the angles of the rhombus.
We know that all the sides of the rhombus are equal in length.
Let the sides AB = BC = CD = AD = x
The altitude from D bisects AB at L.
So, AL = x/2 and LB = x/2
Join DB
Considering triangles ALD and BLD,
As DL is the perpendicular bisector of AB
∠DLA = ∠DLB = 90
Also, AL = BL = x/2
Common side = DL
SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are congruent.
By SAS criteria, the triangles ALD and BLD are congruent.
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCTC,
AD = BD
Now, in triangle ADB AD = AB = DB = x
Therefore, ADB is an equilateral triangle
We know that the angles of an equilateral triangle is always equal to 60 degrees
So, ∠A = ∠ADB = ∠ABD = 60º -------------- (1)
Considering triangle DBC,
DB = BC = CD = x
Therefore, DBC is an equilateral triangle
So, ∠C = ∠CBD = ∠DBC = 60º ------------------- (2)
From (1) and (2), ∠A = ∠C
∠C + ∠B + ∠D = 180º
60 + ∠B + ∠D = 180º
We know, ∠B = ∠D
∠B = ∠D = 180 - 60º
∠B = ∠D = 120º
Therefore, the angles of the rhombus are ∠A = ∠C = 60º and ∠B = ∠D = 120º
✦ Try This: If E,F,G and H are respectively the mid-points of the sides of a parallelogram ABCD, show that ar(EFGH) = 1/2 ar(ABCD)
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.3 Problem 4
ABCD is a rhombus in which altitude from D to side AB bisects AB. Find the angles of the rhombus.
Summary:
A rhombus is a quadrilateral whose four sides all have the same length. ABCD is a rhombus in which altitude from D to side AB bisects AB. The angles of the rhombus are 60º, 120º, 60º and 120º
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