Question

Problem 7.

Let ABCD be points lying on a circle, in this order. Show that the opposite angles of
the quadrilateral ABCD add up to 180°.

Answer 1

A + C = 111.7° +68.3° = 180°

B + D = 45.6° +134.4° = 180°

Explanation:

Apparently, we're to add opposite angles. Your basic math skills (or your calculator) will show you their sum is 180°.

A + C = 111.7° +68.3° = 180°

B + D = 45.6° +134.4° = 180°

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The relationship between an arc and the inscribed angle intercepting it is that the measure of the angle is half the measure of the arc.

Opposite angles of an inscribed quadrilateral intercept the same two points on the circle. The long arc and the short arc together make the whole circle, so the sum of the two angle measures will be half the measure of the whole circle:

360°/2 = 180°

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For example, Angle A intercepts long arc BCD, so is half its measure:

∠A = (1/2)BCD

Its opposite, Angle C, intercepts short arc BD, so is half its measure:

∠C = (1/2)BD

The sum of the two angles is then ...

∠A +∠C = (1/2)BCD + (1/2)BD

= (1/2)(BCD +BD)

= (1/2)360°

= 180°