Question Video: Finding the Measure of an Angle Given Its Minor and Major Arcs’ Measures Mathematics • First Year of Secondary School

Video Transcript

Find 𝑥.

Considering our image, we have a circle and we have two segments that intersect at a point exterior to the circle. Let’s consider what other information we can determine based on what we’re given. We see that angle 𝐵𝐸𝐷 measures 30 degrees and the intercepted arc of this angle is 𝐵𝐷. Additionally, we should notice that angle 𝐵𝐶𝐷 also shares the intercepted arc 𝐵𝐷. And if two angles have the same intercepted arc, they’ll have the same angle measure. Therefore, the measure of angle 𝐵𝐶𝐷 is also 30 degrees.

From there, we can look at triangle 𝐴𝐶𝐷. In this triangle, we have an angle that measures 30 degrees and an angle that measures 40 degrees. And therefore the measure of the missing angle 𝐶𝐷𝐴 will be equal to 180 degrees minus 30 degrees plus 40 degrees. We know the sum of these three angles must be 180 degrees since they form a triangle. Therefore, the measure of angle 𝐶𝐷𝐴 must be 110 degrees.

If we then consider triangle 𝐴𝐵𝐸, we know the measures of two of the angles in the triangle, one being 30 degrees and the other being 40 degrees. Then the missing third angle, the measure of angle 𝐴𝐵𝐸, will also be equal to 180 degrees minus 30 degrees plus 40 degrees, which is 110 degrees.

If we let the intersection of segments 𝐶𝐷 and 𝐵𝐸 be 𝑀, we can consider the quadrilateral 𝑀𝐷𝐴𝐵. And the measure of angle 𝐵𝑀𝐷 will be equal to 360 degrees minus 110 degrees plus 110 degrees plus 40 degrees. This is because in a quadrilateral the interior angles must sum to 360 degrees. The measure of angle 𝐵𝑀𝐷 equals 100 degrees.

For intersecting lines, opposite angles are congruent. The measure of angle 𝐶𝑀𝐸 in the diagram is 100 degrees, which makes 𝑥 equal to 100.