In triangle xyz, if angle x is congruent to angle z, xy = 13x - 21, yz = 8x - 6, & xz = x + 4, find x & the measure of each angle.

Given: In triangle XYZ, , XY=13x-21 , YZ=8x-6 and XZ=x+4.

If two angles are congruent if and only if, they measure the same  number of degrees.

therefore, from the given condition;  .

Isosceles Triangle : An triangle with two equal sides. The angles opposite the equal sides are also equal.

Therefore, the triangle XYZ is an isosceles triangle, as the base angle and are equal and the sides opposite to these angles are also equal i.e,  XY=YZ.

Since, we have XY=YZ ,

⇒ 13x-21 =8x-6 or

13x-8x=21-6   or

5x=15

Simplify:

x=3

Therefore, the sides of an isosceles triangle are:

XY = 13x-21 = = 39-21 = 18 unit ,

YZ = 8x-6 = = 24-6 =18 unit,

and XZ = x+4 = 3+4 =7 unit.

Since, the triangle is isosceles, we draw a line from an vertex angle Y of a triangle XYZ which is perpendicular (i.e, of 90 degree angle) to opposite side meets the opposite side at its midpoint i.e, say W.

As W is the midpoint(i.e, it divide the sides into two equal halves),

XW=WZ = unit

Now, use the Pythagorean theorem, to find the altitude WY.

⇒

Substitute the value of WZ = 3.5 unit and YZ = 18 unit in above formula to calculate the length of WY;

⇒

Simplify:

or

On simplify we get;

WY=17.65 unit(approx.)

The vertex angle Y is split into two equal angles, we can find the vertex angle by finding the one of the base angle by using the fact; .

⇒

⇒ (approx) or  

Therefore, the angle .

The sum of the measure of the angle in the triangle is 180 degree.

In triangle XYZ.

Since

then,

⇒

Simplify:

.

Therefore, the measure of each angle are: and

hello :

by applying the distributive property :

6 × 4.3 = 6 × ( 4 + 0.3 )

            =   (6 ×   4) + (6 ×   0.3)

            = 24 + 1.8

6 × 4.3 = 25.8