Question

In this diagram,ray EA is perpendicular to ray ED and ray EB bisects angle AEC.Given that the measure of angle AEB=4x+1 and the measure of angle CED=3x,find x and the measure of angle BEC.

Answer 1

Answer:  x=8

 ∠BEC =33°  

Step-by-step explanation:

Given : Ray EA is perpendicular to ray ED .

It means ∠AED= 90°  [Perpendicular amkes right angles with the line]  (1)

Also, Ray EB bisects angle AEC.

i.e. ∠AEB=∠BEC

Given that the measure of ∠AEB=4x+1  , then  ∠BEC =4x+1   ......(2)

The measure of ∠CED=3x.                               (3)

Now, ∠AED=∠AEB+∠BEC+∠CED   [From the figure below]

$\Rightarrow\ 90=4x+1+4x+1+3x\ \ \text{[Using (1), (2) and (3)]}\\\\\Rightarrow\ 90=11x+2\\\\\Rightarrow\ 11x=90-2=88\\\\\Rightarrow\ x=\dfrac{88}{11}=8$

Then, ∠BEC =4(8)+1=33°  

Answer 2

If ray EB bisects angle AEC, then m∠AEB=m∠BEC=4x+1.

Angles AEB, BEC and CED together form right angle AED, then

m∠AEB+m∠BEC+m∠CED=90°.

Therefore,

(4x+1)+(4x+1)+3x=90,

11x=90-2,

11x=88,

x=88:11,

x=8°.

Then m∠BEC=4x+1=4·8+1=33°.

Answer: x=8°,  m∠BEC=33°.