ISEE Upper Level Quantitative : Plane Geometry
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Example Questions
Example Question #1 : How To Find An Angle
In isosceles triangle ABC, the measure of angle A is 50 degrees. Which is NOT a possible measure for angle B?
65 degrees
95 degrees
50 degrees
80 degrees
95 degrees
If angle A is one of the base angles, then the other base angle must measure 50 degrees. Since 50 + 50 + x = 180 means x = 80, the vertex angle must measure 80 degrees.
If angle A is the vertex angle, the two base angles must be equal. Since 50 + x + x = 180 means x = 65, the two base angles must measure 65 degrees.
The only number given that is not possible is 95 degrees.
Example Question #2 : How To Find An Angle
The angles of a triangle measure 
The sum of the measures of three angles of a triangle is 
We can simplify and solve for 
Example Question #3 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Let the three angles of a triangle measure 
Which of the following expressions is equal to 
The sum of the measures of the angles of a triangle is 
Example Question #3 : How To Find An Angle
Which of the following is true about a triangle with two angles that measure 
The triangle is acute and scalene.
The triangle is obtuse and scalene.
The triangle cannot exist.
The triangle is obtuse and isosceles.
The triangle is acute and isosceles.
The triangle is obtuse and isosceles.
The measures of the angles of a triangle total 
This makes the triangle obtuse.
Also, since the triangle has two congruent angles (the 
Example Question #4 : How To Find An Angle
You are given two triangles, 
Which quantity is greater?
(a)
(b)
(b) is greater
(a) and (b) are equal
(a) is greater
It is impossible to tell from the information given
(b) is greater
We invoke the SAS Inequality Theorem, which states that, given two triangles 
Example Question #3 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Note: Figure NOT drawn to scale.
Refer to the above figure. Which is the greater quantity?
(a)
(b)
(a) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(b) is greater.
(a) is greater.
(a) The measures of the angles of a linear pair total 180, so:
(b) The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore, 
Therefore (a) is the greater quantity.
Example Question #4 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Note: Figure NOT drawn to scale.
Refer to the above figure. Which is the greater quantity?
(a)
(b)
(b) is greater.
(a) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(a) and (b) are equal.
The two angles at bottom are marked as congruent. Each of these two angles forms a linear pair with a 
The sum of the measures of the interior angles of a triangle is 
The quantities are equal.
Example Question #1 : Geometry
Refer to the above figure. Which is the greater quantity?
(a)
(b)
(b) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(a) is greater.
(a) and (b) are equal.
The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore,
making the quantities equal.
Example Question #1 : Acute / Obtuse Triangles
Which is the greater quantity?
(a)
(b)
(b) is greater.
(a) and (b) are equal.
(a) is greater.
It is impossible to tell from the information given.
(a) is greater.
In 
Since the triangles have two pair of congruent sides, the third side with the greater length is opposite the angle of greater measure. Therefore,
Since 
Example Question #4 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
Which is the greater quantity?
(a)
(b)
(a) and (b) are equal
(b) is the greater quantity
It cannot be determined which of (a) and (b) is greater
(a) is the greater quantity
(a) and (b) are equal
Corresponding angles of similar triangles are congruent, so, since 
By similarity, 
Also,
and 
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