ISEE Upper Level Math : Lines

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

Example Question #2041 : Hspt Mathematics

Vertical_angles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the measure of .

Possible Answers:

Correct answer:

Explanation:

The top and bottom angles, being vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure, so 

or, simplified,

The right and bottom angles form a linear pair, so their degree measures total 180. That is, 

Substitute  for :

The left and right angles, being vertical angles, have the same measure, so, since the right angle measures , this is also the measure of the left angle, .

Example Question #41 : Plane Geometry

Thingy

Figure NOT drawn to scale

The above figure shows Trapezoid , with  and  tangent to the circle. ; evaluate .

Possible Answers:

Correct answer:

Explanation:

By the Same-Side Interior Angle Theorem, since  and  are supplementary - that is, their degree measures total . Therefore, 

 is an inscribed angle, so the arc it intercepts, , has twice its degree measure;

.

The corresponding major arc, , has as its measure 

The measure of an angle formed by two tangents to a circle is equal to half the difference of those of its intercepted arcs:

Again, by the Same-Side Interior Angles Theorem,  and  are supplementary, so

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