ISEE Upper Level

Angles

Angle relationships (supplementary, complementary, vertical), parallel line angles, and interior angle sums of polygons

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Imagine you are slicing a giant, delicious pizza. Every cut you make creates angles and lines! 🍕 In geometry, angles are everywhere. Think of a door opening—the wider it opens, the bigger the angle. On the ISEE, you will be a geometry detective looking for clues hidden in lines and shapes.

We have special names for these math clues. For instance, a perfectly straight line is always $18 0^{\circ}$ . If you see a straight line split into two angles, they are called "supplementary" (which just means they add up to 180, like a super-sized skateboard trick!). If two lines cross like a giant "X", the angles opposite each other are equal, like twins! 👯

You will also see shapes like triangles and squares. The inside angles of any triangle always add up to exactly $18 0^{\circ}$ . A square's corners are always $9 0^{\circ}$ (perfect right angles, just like the corner of your favorite book).

The ISEE loves to test these rules in both of its math sections. Sometimes you will calculate a missing angle, and other times you will use the Quantitative Comparison format to see which angle is bigger. Remember, you don't need a protractor for this test—just use your trusty math rules and you will do great! 📐✨

Key Formula

The "C and S" Rule: Complementary angles add to

9 0^{\circ}

and Supplementary angles add to

18 0^{\circ}

. (Trick: "C" comes before "S" in the alphabet, and

90

comes before

180

in numbers!)

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

Angle

A

and angle

B

are supplementary. Angle

A

12

degrees less than twice the measure of angle

B

. What is the measure of angle

A

64

degrees

108

degrees

116

degrees

124

degrees

Show Solution

Let $A$ be the measure of angle $A$ and $B$ be the measure of angle $B$ .
1. Understand 'supplementary': Two angles are supplementary if their sum is $180$ degrees. So, $A + B = 180$ .
2. Translate the relationship: "Angle $A$ is $12$ degrees less than twice the measure of angle $B$ " can be written as $A = 2 B - 12$ .
3. Solve the system of equations: Substitute the expression for $A$ from the second equation into the first equation:

$(2 B - 12) + B = 180$
$3 B - 12 = 180$
$3 B = 180 + 12$
$3 B = 192$
$B = \frac{192}{3}$
$B = 64$ degrees
4. Find angle $A$ : Now substitute the value of $B$ back into the equation $A = 2 B - 12$ :

$A = 2 (64) - 12$
$A = 128 - 12$
$A = 116$ degrees
To verify, $116 + 64 = 180$ , which confirms they are supplementary.

Answer: C

Q2 Hard

Lines

L

and

M

are parallel and are intersected by transversal

T

. Let angle 1 and angle 2 be alternate interior angles. If angle 1 measures

(4 x - 15)

degrees and angle 2 measures

(2 x + 35)

degrees, what is the measure of the angle vertically opposite to angle 1?

55

degrees

75

degrees

85

degrees

95

degrees

Show Solution

1. Identify the relationship: Alternate interior angles are equal when two parallel lines are cut by a transversal.

Therefore, $an g l e 1 = an g l e 2$ .
2. Set up and solve for $x$ : Substitute the given expressions for angle 1 and angle 2:

$4 x - 15 = 2 x + 35$
Subtract $2 x$ from both sides: $2 x - 15 = 35$
Add $15$ to both sides: $2 x = 50$
Divide by $2$ : $x = 25$
3. Calculate the measure of angle 1: Substitute the value of $x$ back into the expression for angle 1:

$an g l e 1 = 4 (25) - 15$
$an g l e 1 = 100 - 15$
$an g l e 1 = 85$ degrees
4. Find the vertically opposite angle: Vertically opposite angles are equal. The angle vertically opposite to angle 1 will have the same measure as angle 1.

Therefore, the measure of the angle vertically opposite to angle 1 is $85$ degrees.

Answer: C

Q3 Hard

A convex hexagon has five of its interior angles measuring

110

degrees,

125

degrees,

130

degrees,

140

degrees, and

155

degrees. What is the measure of the sixth interior angle?

60

degrees

70

degrees

85

degrees

95

degrees

Show Solution

1. Find the sum of interior angles of a hexagon: The formula for the sum of the interior angles of a convex polygon with $n$ sides is $(n - 2) \times 180$ degrees.

For a hexagon, $n = 6$ .
Sum of angles = $(6 - 2) \times 180 = 4 \times 180 = 720$ degrees.
2. Sum the given angles: Add the measures of the five known interior angles:

$110 + 125 + 130 + 140 + 155 = 660$ degrees.
3. Calculate the sixth angle: Subtract the sum of the five angles from the total sum of angles for a hexagon:

Sixth angle = $720 - 660 = 60$ degrees.

Answer: A

Q4 Hard

Lines

L

and

M

are parallel. Transversal

T

intersects

L

at point

A

and

M

at point

B

. A second transversal,

S

, intersects

L

at point

C

and

M

at point

D

. If angle CAB measures

130

degrees and angle ADB measures

35

degrees, what is the measure of angle ACD?

40

degrees

50

degrees

65

degrees

75

degrees

Show Solution

Let's visualize the setup. Points $C$ and $A$ are on line $L$ , and points $D$ and $B$ are on line $M$ . $L$ is parallel to $M$ . $A B$ and $C D$ are transversals.

We are looking for angle ACD, which is an angle in the triangle $A C D$ .
1. Find angle ABD: angle CAB and angle ABD are consecutive interior angles formed by parallel lines $L$ and $M$ and transversal $A B$ . Consecutive interior angles are supplementary.

$an g l e C A B + an g l e A B D = 180$
$130 + an g l e A B D = 180$
$an g l e A B D = 180 - 130 = 50$ degrees.
2. Find angle BAD: Consider triangle $A B D$ . The sum of angles in a triangle is $180$ degrees.

We have $an g l e A D B = 35$ degrees (given) and $an g l e A B D = 50$ degrees (calculated).
$an g l e B A D + an g l e A D B + an g l e A B D = 180$
$an g l e B A D + 35 + 50 = 180$
$an g l e B A D + 85 = 180$
$an g l e B A D = 180 - 85 = 95$ degrees.
3. Identify angles for triangle $A C D$ : The angle angle CAD in triangle $A C D$ is the same as angle BAD, as $C$ , $A$ are on line $L$ and $D$ is on line $M$ . So $an g l e C A D = 95$ degrees.

The angle angle ADC in triangle $A C D$ is the same as the given angle ADB, as $B$ , $D$ are on line $M$ and $A$ is on line $L$ . So $an g l e A D C = 35$ degrees.
4. Find angle ACD: In triangle $A C D$ , the sum of angles is $180$ degrees.

$an g l e A C D + an g l e C A D + an g l e A D C = 180$
$an g l e A C D + 95 + 35 = 180$
$an g l e A C D + 130 = 180$
$an g l e A C D = 180 - 130 = 50$ degrees.

Answer: B

Tips & Strategies

Memorize the 'C and S' trick! Complementary angles add to 90° and Supplementary angles add to 180°. Alphabetical order matches numerical order!
In Quantitative Comparison questions, don't trust your eyes! The drawings on the ISEE are often NOT drawn to scale. An angle might look like 90° but actually be 89°. Always use the numbers given in the text.

Common Mistakes

Watch out for mixing up complementary and supplementary angles. If the question asks for a complement, don't accidentally subtract the number from 180°!
Don't forget that angles around a single center point form a full circle. That means all the angles touching that center point must add up to exactly 360°.

Frequently Asked Questions

Do I need to bring a protractor to the ISEE?

Nope! Protractors aren't allowed on the test. The ISEE wants you to use math rules (like knowing a straight line is 180°) to calculate missing angles, not a measuring tool.

What is a 'transversal' line?

A transversal is just a fancy geometry word for a line that crashes through two or more other lines. When it cuts through parallel lines, it creates a super cool pattern of matching angles!

What if I get stuck on a Quantitative Comparison angle question?

If you are totally stuck, try to eliminate choices. Also, if the angles are just regular numbers, the answer is rarely (D) 'Cannot be determined'. Make your best guess, because there is NO penalty for guessing on the ISEE!

What are 'vertical angles'?

When two straight lines cross each other like a giant 'X', the angles directly across from each other are called vertical angles. They are always perfectly equal to each other!

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