Question

you do: check for understanding
find the value of m.
a. 3
b. 4
c. 5
d. 6

you do: check for understanding

1. in an equilateral triangle, what is the difference between the sum of the exterior and interior angles?
2. find the measure of angle c.

Explanation:

First Problem: Find the value of $m$

Step1: Identify equal sides/angles

The triangle has two $59^\circ$ angles, so the sides opposite them are equal: $11m + 5 = 16m - 20$

Step2: Isolate $m$ terms

Subtract $11m$ from both sides: $5 = 5m - 20$

Step3: Solve for $m$

Add 20 to both sides, then divide by 5:
$25 = 5m$
$m = \frac{25}{5} = 5$

The sum of interior angles of any triangle is $180^\circ$. The sum of exterior angles of any polygon (including triangles) is always $360^\circ$. Calculate the difference between these two values.

Step1: Set equal side expressions

Sides $AB$ and $AC$ are equal, so $6b - 8 = 4b + 16$

Step2: Solve for $b$

Subtract $4b$, add 8:
$2b = 24$
$b = 12$

Step3: Solve for $a$ using angle sum

Sum of triangle angles is $180^\circ$:
$(3a + 26) + (15a - 82) + \angle C = 180^\circ$
Since it's isosceles, $\angle B = \angle C = 3a + 26$. Substitute:
$(3a + 26) + (15a - 82) + (3a + 26) = 180$
Combine like terms: $21a - 30 = 180$

Step4: Calculate $a$

$21a = 210$
$a = 10$

Step5: Find $\angle C$

Substitute $a=10$: $\angle C = 3(10) + 26 = 56$

Answer:

c. 5

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Second Problem (1): Difference between sum of exterior and interior angles of equilateral triangle