Help..............................
1 answer:
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Answer:
The measure of arc ABC is 180 + x ⇒ 1st answer
Step-by-step explanation:
Let us revise some notes about the arcs of a circle
- The measure of an inscribed angle is half the measure of its subtended arc
- The measure of the central angle is equal to the measure of its subtended arc
- If two tangents intersect outside a circle then the measure of the angle formed between them is one-half the positive difference of the measures of the intercepted arcs
∵ CD and AD are two tangents to the given circle
∵ They intersected at point D out side the circle
∵ The measure of ∠ADC = x°
- By using the 3rd note above
∵ The intercepted arcs are arc ABC and arc AC
∴ x° = (measure of arc ABC - measure of arc AC)
- Multiply both sides by 2
∴ 2 x = (measure of arc ABC - measure of arc AC)
- Add measure of arc AC in both sides
∴ 2 x + measure of arc AC = measure of arc ABC ⇒ (1)
∵ The measure of a circle is 360°
∵ The arcs ARC and AC is formed the circle
∴ Measure of arc ABC + measure of arc AC = 360°
- Subtract measure of arc ABC from both sides
∵ Measure of arc AC = 360 - measure of arc ABC ⇒ (2)
- Substitute (2) in (1)
∴ 2 x + 360 - measure of arc ABC = measure of arc ABC
- Add measure of arc ABC to both sides
∴ 2 x + 360 = 2 measure of arc ABC
- Divide both sides by 2
∴ x + 180 = measure of arc ABC
∴ The measure of arc ABC is 180 + x
Since you did not provide a diagram, I made one for easier reference. You are asking what kind of triangle is the one shaded in red. Since a rectangle has two sets of congruent sides, their diagonals are also equal. When two diagonals intersect each other, they make 4 congruent sides. Thus, the triangle is an isosceles triangle, since it has two equal sides ST and TR.
