Question

1) Given: mLHE=84°

Find: m∠EYL.

2)Given: m∠EYL=72°
Find: m arc EHL, m arc LVE

Answer 1

Answer:

Part 1) The measure of angle EYL is $96\°$

Part 2) The measure of arc EHL is $108\°$ and the measure of arc LVE is $252\°$

Step-by-step explanation:

we know that

The measurement of the outer angle is the semi-difference of the arcs which comprises

Part 1)

Let

x------> the measure of arc LHE

y----> the measure of arc LVE

we know that

$x+y=360\°$

$x=84\°$

Find the value of y

$y=360\°-84\°=276\°$

Find the measure of angle EYL

$m<EYL=\frac{1}{2} (y-x)$

substitute the values

$m<EYL=\frac{1}{2}(276\°-84\°)=96\°$

Part 2)

Let

x------> the measure of arc EHL

y----> the measure of arc LVE

we know that

$x+y=360\°$

$x=360\°-y$ -----> equation A

$m<EYL=72\°$

$m<EYL=\frac{1}{2} (y-x)$

substitute

$72\°=\frac{1}{2} (y-x)$

$144\°=(y-x)$

$x=y-144\°$ --------> equation B

equate equation A and equation B and solve for y

$360\°-y=y-144\°$

$2y=360\°+144\°$

$2y=504\°$

$y=252\°$

Find the value of x

$x=252\°-144\°=108\°$

therefore

The measure of arc EHL is $108\°$

The measure of arc LVE is $252\°$

Answer 2

A circle is a curve sketched out by a point moving in a plane. The measure of the arcEHL and arcLVE are 108° and 252° respectively.

What is a circle?

A circle is a curve sketched out by a point moving in a plane so that its distance from a given point is constant; alternatively, it is the shape formed by all points in a plane that are at a set distance from a given point, the centre.

As we can see in the image attached below, the radius of the circle is OE and OL while there is two tangent to the circle Ey and LY. Therefore, the measure of the ∠OEY and ∠OLY is 90°.

A.)

The sum of the angles of a quadrilateral is 360°.

As it is mentioned in the problem the measure of the angle made by arc LHE is 84° which means the measure of ∠EOL is 84°. Therefore, the sum of all the angles can be written as,

$\text{Sum of angles} = 360^o\\\\\angle OEY + \angle OLY + \angle EOL + \angle EYL = 360^o\\\\90^o+90^o+84^o +\angle EYL = 360^o\\\\\angle EYL = 96^o$

B.)

The sum of the angles of a quadrilateral is 360°.

As it is mentioned in the problem the measure of ∠EYL is 72°. Therefore, the sum of all the angles can be written as,

$\text{Sum of angles} = 360^o\\\\\angle OEY + \angle OLY + \angle EOL + \angle EYL = 360^o\\\\90^o+90^o+ \angle EOL +72^o = 360^o\\\\ \angle EOL= 108^o\\\\ \rm arc EHL=108^o$

Since a complete circle measures 360°, therefore, the sum of the angles made by arc EHL and arc LVE can be written as,

arcEHL + arcLVE = 360°

108° + arcLVE = 360°

arcLVE = 252°

Thus, the measure of the arcEHL and arcLVE are 108° and 252° respectively.

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