SSAT Upper Level

Triangles

Triangle area, Pythagorean theorem, angle sums, congruence, and similarity — excludes general area/perimeter of other shapes (see area-perimeter-composite)

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Imagine grabbing a perfect slice of pizza. It’s not just delicious; it’s the best shape ever—a triangle! Triangles are secretly hiding everywhere in our world, from sandwich halves in your lunchbox to giant skateboard ramps and even the ancient pyramids in Egypt. 🍕

On the SSAT, triangles are definitely VIPs (Very Important Polygons). But what makes a triangle a triangle? Every triangle has three straight sides and three inside angles. Here is the golden rule: those three inside angles always, always, always add up to exactly $18 0^{\circ}$ . Think of it as a strict dress code at the Triangle Club. If your angles add up to $17 9^{\circ}$ or $18 1^{\circ}$ , the bouncer won't let you in! 📐

There’s another wild rule you need to know called the Triangle Inequality Theorem. It says that if you add the lengths of the two shortest sides of a triangle, the total must be bigger than the longest side. If they aren't, the sides won't be able to reach each other to close the shape, kind of like a drawbridge that falls short over a castle moat. Finally, keep an eye out for "Right Triangles" (triangles with a perfect $9 0^{\circ}$ corner, like the corner of a book). They use a magic spell called the Pythagorean Theorem to find missing sides. Master these fun shapes, and you'll completely crush the geometry questions on your SSAT! 🦸‍♂️

Key Formula

For a right triangle with short legs

a

and

b

, and the longest side (hypotenuse)

c

, the magic formula is the Pythagorean Theorem:

a^{2} + b^{2} = c^{2}

. Also, for finding the area of any triangle, always use

Area = \frac{1}{2} \cdot b \cdot h

Practice Questions

5 practice questions for SSAT Upper Level

Q1 Hard

△ P QR

, side

\overline{QR}

is extended through point

S

to form an exterior angle

∠ P R S

. If the measure of

∠ P QR

4 5^{\circ}

and the measure of

∠ QP R

6 5^{\circ}

, what is the measure of

∠ P R S

2 0^{\circ}

7 0^{\circ}

11 0^{\circ}

12 0^{\circ}

14 5^{\circ}

Show Solution

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote (opposite) interior angles. Therefore, the measure of $∠ P R S$ is equal to the sum of $∠ P QR$ and $∠ QP R$ . Adding these together gives $4 5^{\circ} + 6 5^{\circ} = 11 0^{\circ}$ . Alternatively, you can find the third interior angle, $∠ P R Q = 18 0^{\circ} - (4 5^{\circ} + 6 5^{\circ}) = 7 0^{\circ}$ , and then find its supplementary exterior angle: $18 0^{\circ} - 7 0^{\circ} = 11 0^{\circ}$ .

Answer: C

Q2 Hard

An equilateral triangle has a side length of

8

. What is the height (altitude) of this triangle?

4

42

43

82

83

Show Solution

Drawing an altitude in an equilateral triangle bisects the base and splits the triangle into two 30-60-90 right triangles. The hypotenuse of one of these right triangles is the original side length of $8$ . The shorter leg (half of the base) is $\frac{8}{2} = 4$ . In a 30-60-90 triangle, the longer leg (the altitude) is equal to the shorter leg multiplied by $3$ . Therefore, the height is $43$ .

Answer: C

Q3 Hard

A ladder is leaning against a vertical wall. The base of the ladder is placed

5

feet away from the bottom of the wall. If the top of the ladder reaches exactly

12

feet up the wall, how long is the ladder?

13

feet

14

feet

15

feet

17

feet

19

feet

Show Solution

The wall, the ground, and the ladder form a right triangle. The distance from the wall to the base of the ladder is one leg ( $a = 5$ ), and the height the ladder reaches on the wall is the second leg ( $b = 12$ ). The length of the ladder is the hypotenuse ( $c$ ). Using the Pythagorean theorem, $a^{2} + b^{2} = c^{2}$ , we get $5^{2} + 1 2^{2} = c^{2}$ . This simplifies to $25 + 144 = 169 = c^{2}$ . Taking the square root of both sides gives $c = 13$ . The ladder is $13$ feet long.

Answer: A

Q4 Hard

In quadrilateral

W X Y Z

∠ W X Y

and

∠ W Y Z

are both right angles. If

W X = 6

X Y = 8

, and

Y Z = 24

, what is the length of

\overline{W Z}

10

18

24

26

30

Show Solution

This problem involves two connected right triangles. First, look at right triangle $△ W X Y$ . The legs are $W X = 6$ and $X Y = 8$ . Using the Pythagorean theorem, the hypotenuse $W Y$ is $6^{2} + 8^{2} = 36 + 64 = 100 = 10$ . Next, look at right triangle $△ W Y Z$ . The legs are the newly found $W Y = 10$ and the given $Y Z = 24$ . Using the Pythagorean theorem again, the hypotenuse $W Z$ is $1 0^{2} + 2 4^{2} = 100 + 576 = 676 = 26$ .

Answer: D

Q5 Hard

△ D E F

\overline{D E}

and

\overline{D F}

are equal in length. If the measure of

∠ D

5 0^{\circ}

, what is the measure of

∠ E

5 0^{\circ}

6 5^{\circ}

8 0^{\circ}

10 0^{\circ}

13 0^{\circ}

Show Solution

Because sides $\overline{D E}$ and $\overline{D F}$ are equal, $△ D E F$ is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are also equal, meaning $∠ E = ∠ F$ . The sum of the interior angles of any triangle is $18 0^{\circ}$ . Subtracting the known angle $∠ D$ gives $18 0^{\circ} - 5 0^{\circ} = 13 0^{\circ}$ . This remaining degree measure is split equally between $∠ E$ and $∠ F$ . Thus, the measure of $∠ E$ is $\frac{13 0 ^{\circ}}{2} = 6 5^{\circ}$ .

Answer: B

Tips & Strategies

Memorize common 'Pythagorean Triples' like $3 - 4 - 5$ and $5 - 12 - 13$ . If you see a right triangle with legs $3$ and $4$ , you instantly know the hypotenuse is $5$ without doing any math! ⚡
If an SSAT question asks about the third side of a triangle, always check the Triangle Inequality rule. The third side must be smaller than the sum of the other two sides, and larger than their difference.
Draw it out! If the test describes a triangle but doesn't give you a picture, sketch it on your scratch paper. Seeing the shape makes finding the missing angles or sides much easier. ✏️

Common Mistakes

🚨 Watch out for assuming a triangle is a right triangle just because it looks like one! Unless the problem puts a little square in the corner (the $9 0^{\circ}$ symbol) or tells you it's a right triangle, you cannot use $a^{2} + b^{2} = c^{2}$ .
Don't forget that the hypotenuse is ALWAYS the longest side. If you calculate $c$ and it's smaller than $a$ or $b$ , double-check your math!

Frequently Asked Questions

Do I need to memorize area formulas for triangles on the SSAT?

Yes! The area of a triangle is very important. Always remember the formula $Area = \frac{1}{2} \cdot b \cdot h$ , where $b$ is the base and $h$ is the height.

What does 'congruent' mean?

Congruent is just a fancy math word for 'exactly the same.' If two triangles are congruent, their sides and angles are perfectly identical, like twins!

Will the SSAT give me the formulas I need?

Nope! Unlike some other tests, the SSAT does not give you a formula sheet. You'll need to memorize important rules like the Pythagorean Theorem and the $18 0^{\circ}$ angle rule before test day.

What is a 'similar' triangle?

Similar triangles have the exact same shape, but different sizes. Think of it like zooming in or out on a picture on a smartphone. Their angles match perfectly, and their sides grow or shrink by the same fraction.

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