Question

for what value of x will make the two triangles similar? image of two triangles options: x = 17; x = 0.7; x = 10.2; x = 5.3. question 4 2 points given the following angles of the quadrilateral, find the value of angle 4. m∠1 = (10x - 5)°; m∠2 = (12x - 6)°; m∠3 = (11x + 2)°; m∠4 = (14x - 7)° options: 75°; 105°; 8°; 90°

Explanation:

First Problem (Similar Triangles)

Step1: Set up proportion of sides

For similar triangles, corresponding sides are proportional: $\frac{7}{2} = \frac{x+8}{x-5}$

Step2: Cross-multiply to eliminate fractions

$7(x-5) = 2(x+8)$

Step3: Expand both sides

$7x - 35 = 2x + 16$

Step4: Isolate x terms

$7x - 2x = 16 + 35$

Step5: Calculate x value

$5x = 51 \implies x = \frac{51}{5} = 10.2$

Step1: Sum of quadrilateral angles

The sum of interior angles of a quadrilateral is $360^\circ$:
$(10x-5)+(12x-6)+(11x+2)+(14x-7) = 360$

Step2: Combine like terms

$10x+12x+11x+14x -5-6+2-7 = 360$
$47x - 16 = 360$

Step3: Solve for x

$47x = 360 + 16 = 376 \implies x = \frac{376}{47} = 8$

Step4: Calculate angle 4

Substitute $x=8$ into $m\angle4=(14x-7)^\circ$:
$m\angle4 = 14(8)-7 = 112 -7 = 105^\circ$

Answer:

x = 10.2

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Second Problem (Quadrilateral Angles)