Answer:
12. m∠C = 100° and m∠F = 100°
13. m∠S = 89° and m∠U = 89°
Explanation:
Question 12
The two triangles ΔABC and ΔDEF have two of their angles equal:
m ∠A = m ∠D
m∠B = m∠E
Therefore the third angles must be equal:
m∠C = m∠F
m∠C = (4x²)° and m∠F = (3x² + 25)°
Setting these two right side expressions in parentheses equal to each other gives
4x² = 3x² + 25
4x² - 3x² = 25
x² = 25
x = ±√25 = ±5
Ignoring the negative value we get
x = 5
Therefore
m∠C = (4x²)° = (4 · 5²)° = (4 · 25)° = 100°
Since, m∠F = m∠C ,
m∠F = 100°
But we can double check
m∠F = (3x² + 25)° = (3 · 5² + 25)° = (3 · 25 + 25) = (75 + 25)° = 100°
Question 13
This is similar to question 12
The quadrilateral RSTU is divided into two triangles ΔRST and Δ RUT
We are given
m∠RTS = m∠TRU
m∠TRS = m∠RTU
Therefore the third angle of ΔRST must be equal to the third angle of Δ RUT
In other words,
m∠S= m∠U
Plugging in the expressions for each of these angles we get
(5x - 11)° = (4x + 9)°
Set the expressions inside parens equal to each other and solve for x:
5x - 11 = 4x + 9
5x - 4x - 11 = 9 (subtract 4x from both sides)
x - 11 = 9
x = 11 + 9 (add 11 both sides)
x = 20
m∠S= (4x + 9)° = (4 · 20 + 9)° = (80 + 9)° = 89°
Therefore m∠RUT is also 89°
but we can double-check
m∠U= (5x - 11)° = (5 · 20 - 11)° = (100 - 11)° = 89°