Question

isosceles triangles
find the values of the variables for each triangle.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.

Explanation:

Step1: Solve Triangle 1

Sum of angles in triangle: $a + 60 + 60 = 180$
$a = 180 - 120 = 60$. Equal angles mean equal sides: $b = 10$.

Step2: Solve Triangle 2

Isosceles triangle: $\angle E = \angle F = 40^\circ$, so $x = 40$.
Sum of angles: $y + 40 + 40 = 180$
$y = 180 - 80 = 100$.

Step3: Solve Triangle 3

Vertical angles: $c = 54$. Isosceles $\triangle AXY$: $\angle A = \angle X = d$.
Sum of angles in $\triangle AXY$: $d + d + 54 = 180$
$2d = 126 \implies d = 61$.

Step4: Solve Triangle 4

Right isosceles triangle: $x = 45$, $y = 4$.

Step5: Solve Triangle 5

Isosceles $\triangle PCA$: $a = 60$. Isosceles $\triangle PMA$: $\angle MPA = \angle MAP$.
Sum of angles in $\triangle PMA$: $b + b + 44 = 180$
$2b = 136 \implies b = 68$.

Step6: Solve Triangle 6

Isosceles $\triangle RAX$: $\angle RAX = \angle R = 40$. Linear pair: $x + 40 = 180$
$x = 140$. Vertical angle: $y = x = 140$.

Step7: Solve Triangle 7

Isosceles $\triangle WXY$: $\angle W = \angle X = b = a$.
Sum of angles in $\triangle WXY$: $a + a + 24 = 180$
$2a = 156 \implies a = 78$, so $b = 78$.

Step8: Solve Triangle 8

Right $\triangle RAT$: $x + 45 + 90 = 180$
$x = 45$. Isosceles $\triangle RST$: $\angle TRS = \angle TSR = 60$.
$\angle YRT = 90 - 60 = 30$, so $y = 30$.

Step9: Solve Triangle 9

Parallelogram: $a = \angle B = 52$. Isosceles $\triangle ZED$: $\angle EZD = \angle EDZ$.
Sum of angles in $\triangle ZED$: $52 + b + b = 180$
$2b = 128 \implies b = 64$.

Step10: Solve Triangle 10

Right isosceles triangle: $x = 90$, $a = 45$.

Step11: Solve Triangle 11

Vertical angles: $c = 180 - 44 - 44 = 92$. Isosceles triangle: $d = 44$.

Step12: Solve Triangle 12

Isosceles triangle: $\angle$ opposite 42 is 42. Sum in quadrilateral: $b + 42 + 88 + 90 = 360$
$b = 360 - 220 = 140$. Isosceles triangle: $a = 180 - 88 - 42 = 50$.

Answer:

1. $a=60$, $b=10$
2. $x=40$, $y=100$
3. $d=61$, $c=54$
4. $x=45$, $y=4$
5. $a=60$, $b=68$
6. $x=140$, $y=140$
7. $a=78$, $b=78$
8. $x=45$, $y=30$
9. $a=52$, $b=64$
10. $x=90$, $a=45$
11. $d=44$, $c=92$
12. $a=50$, $b=140$