Answer:
Given: In triangle XYZ, [tex]\angle X \cong \angle Z[/tex] , 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; [tex]\angle X = \angle Z[/tex].
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 [tex]\angle X[/tex] and [tex]\angle Z[/tex] 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 = [tex]13\cdot 3 -21[/tex] = 39-21 = 18 unit ,
YZ = 8x-6 = [tex]8 \cdot3 -4[/tex] = 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 = [tex]\frac{7}{2} =3.5[/tex] unit
Now, use the Pythagorean theorem, to find the altitude WY.
⇒ [tex](WY)^2+(WZ)^2=(YZ)^2[/tex]
Substitute the value of WZ = 3.5 unit and YZ = 18 unit in above formula to calculate the length of WY;
⇒ [tex](WY)^2+(3.5)^2=(18)^2[/tex]
Simplify:
[tex](WY)^2=324-12.25=311.75[/tex] or
[tex]WY=\sqrt{311.75}[/tex]
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; [tex]Cosine = \frac{Base}{Hypotenuse}[/tex].
⇒ [tex]\cos Z =\frac{3.5}{18}[/tex]
⇒ [tex]\cos Z = 0.195[/tex](approx) or
[tex]Z=\cos^{-1}(0.195)[/tex]
Therefore, the angle [tex]\angle Z = 78.7^{\circ}[/tex].
The sum of the measure of the angle in the triangle is 180 degree.
In triangle XYZ.
[tex]\angle X +\angle Y+\angle Z =180^{\circ}[/tex]
Since [tex]\angle X=\angle Z =78.7^{\circ}[/tex]
then,
[tex]78.7^{\circ}+\angle Y+78.7^{\circ}=180^{\circ}[/tex]
⇒[tex]157.4^{\circ}+\angle Y=180^{\circ}[/tex]
Simplify:
[tex]\angle Y=22.6^{\circ}[/tex].
Therefore, the measure of each angle are: [tex]\angle X=\angle Z=78.7^{\circ}[/tex] and [tex]\angle Y =22.6^{\circ}[/tex]
