We can use this theorem to identify the unknown angles in the given triangle.
By the Triangle Angle Sum Theorem, the sum of the measures of the interior angles in a triangle must be equal to 180^(∘). Let's use this theorem to write an equation for x.
x + x + 52^(∘) = 180^(∘)
Now we can solve this equation for x. Let's do it!
b We have been asked to find x based on the given diagram. Notice that the smaller angles are formed by adding a ray starting from a point on line. Therefore, they add up to a straight angle.
Since the measure of any straight angle is 180^(∘), we can use the following formula to relate the measures of these supplementary angles.
(4x-3^(∘))+(3x+1^(∘)) = 180^(∘)
Let's solve the equation for x.
c For any △ ABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively.
The Law of Sines relates the sine of each angle to the length of its opposite side.
sin A/a=sin B/b=sin C/c
Consider the given triangle.
Notice that we cannot use the Law of Sines yet we only have one angle-side length measure pair. However, since we know two out of the three angle measures of the triangle, we can find the last one using the Triangle Angle Sum Theorem. Let's label the measure of the missing angle y. This allows us to write the following equation for y.
y + 77^(∘) + 31^(∘) = 180^(∘)
Let's solve it!
The final angle of our triangle is 72^(∘). Let's add it to the diagram.
Now we know to pairs of angle measures and their opposite side lengths! This means that we can use the Law of Sines to write an equation for the missing length x.
sin 72^(∘)/8 = sin 77^(∘)/x
Let's solve the equation for x.
