Advertisements
Advertisements
Question
ABCDE is a regular pentagon. The bisector of angle A of the pentagon meets the side CD in point M. Show that ∠AMC = 90°.
Advertisements
Solution
Given: ABCDE is a regular pentagon.
The bisector ∠A of the pentagon meets the side CD at point M.
To prove : ∠AMC = 90°
Proof: We know that the measure of each interior angle of a regular pentagon is 108°.
∠BAM = 
Since, we know that the sum of a quadrilateral is 360°
In quadrilateral ABCM, we have
∠BAM + ∠ABC + ∠BCM + ∠AMC = 360°
54° + 108° + 108° + ∠AMC = 360°
∠AMC = 360° – 270°
∠AMC = 90°
APPEARS IN
RELATED QUESTIONS
How many diagonals does following have?
A convex quadrilateral
The measure of angles of a hexagon are x°, (x − 5)°, (x − 5)°, (2x − 5)°, (2x − 5)°, (2x + 20)°. Find the value of x.
In a convex hexagon, prove that the sum of all interior angle is equal to twice the sum of its exterior angles formed by producing the sides in the same order.
The sum of the interior angles of a polygon is three times the sum of its exterior angles. Determine the number of sided of the polygon.
Complete the following statement by means of one of those given in brackets against each:
f consecutive sides of a parallelogram are equal, then it is necessarily a ..................
If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = k ∠AOB, then find the value of k.
In quadrilateral ABCD, side AB is parallel to side DC. If ∠A : ∠D = 1 : 2 and ∠C : ∠B = 4 : 5
(i) Calculate each angle of the quadrilateral.
(ii) Assign a special name to quadrilateral ABCD
Two adjacent angles of a parallelogram are 70° and 110° respectively. Find the other two angles of it.
Find the angles of a quadrilateral whose angles are in the ratio 1: 4: 5: 2.
How many points are required to form a quadrilateral?
