ISEE Upper Level Quantitative : How to find an angle
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Example Questions
Example Question #11 : Geometry
Refer to the above figure. Which is the greater quantity?
(a)
(b)
(a) and (b) are equal
(b) is the greater quantity
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(a) is the greater quantity
Extend as seen in the figure below:
The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles; specifically,
,
and
However, , so, by substitution,
Example Question #11 : Plane Geometry
Given: . . Which is the greater quantity?
(a)
(b)
(a) and (b) are equal
(b) is the greater quantity
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(b) is the greater quantity
Below is the referenced triangle along with , an equilateral triangle with sides of length 10:
As an angle of an equilateral triangle, has measure . Applying the Side-Side-Side Inequality Theorem, since , , and , it follows that , so .
Also, since , by the Isosceles Triangle Theorem, . Since , and the sum of the measures of the angles of a triangle is , it follows that
Substituting and solving:
.
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