Digital SAT · Math · Unit 4 · Geometry & Trigonometry

Lesson 4.2 · Triangles

On the test: 2–4 questions a test — the most tested shape on the SAT: congruence and similarity, right triangles, areas, and the first trig ratios

Big idea. Nearly every triangle question is one of five moves: the angles add to 180°; equal shapes are proved by a shortcut (SSS, SAS, ASA, AAS); similar shapes multiply every side by one number k (and area by k²); right triangles obey a² + b² = c² and come in two special shapes; and area is ½ · base · height. The reference sheet gives you a few of these — the lab below each one shows you WHY.

🎛 Move it · the angles of a triangle

Move the top and change the height: the three angles always make 180°. Tick isosceles to see equal sides over equal angles.

move the top across140height150

make two sides equal (isosceles)

The three angles always add to 180°: 56.3 + 43.2 + 80.5 = 180.

Exterior angle = the two angles NOT next to it: 136.8 = 56.3 + 80.5 (it and its neighbour make 180, and so do all three inside).

🧑‍🏫 Rule 1 · The angles add to 180°step 1 of 3

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📋 The question · SAT · October US 2024 · Math 800 Book Q33
In $△ F G H$ , the measures of $∠ F$ and $∠ G$ are each $39°$ . What is the measure of $∠ H$ ?
1. A) $78°$
2. B) $102°$
3. C) $129°$
4. D) $141°$
two known
$∠ H = 180 - 39 - 39 = 102$
October US 2024: B.

🧑‍🏫 Rule 2 · Isosceles and the exterior anglestep 1 of 4

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📋 The question · Gen Z Study · isosceles
In an isosceles triangle the angle between the two equal sides measures $40°$ . What is the measure, in degrees, of each of the other two angles?
(type the answer)
equal sides → equal angles
$\frac{180 - 40}{2} = 70$
The two angles at the base are equal.

🎛 Move it · does it close?

Two sides and a third. Make the third side too long or too short and the triangle falls apart.

side a8side b13third side10

It closes — a triangle. The third side must be bigger than the difference and smaller than the sum: 5 < third < 21. At exactly 21 the two sides lie flat on it — no triangle. Also: the longest side faces the largest angle.

🧑‍🏫 Rule 3 · The triangle inequality and the biggest anglestep 1 of 4

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📋 The question · SAT · August US 2024 · Math 800 Book Q30
A triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 8 and 13, which inequality represents the possible length, $x$ , of the third side of the triangle?
1. A) $x < 21$
2. B) $x > 21$
3. C) $5 < x < 21$
4. D) $x < 5$ or $x > 21$
difference < third < sum
$13 - 8 < x < 13 + 8 \Rightarrow 5 < x < 21$
August US 2024: C.

🎛 Move it · the congruence shortcuts

Pick a shortcut. The given parts turn red. SSA: watch the side swing to TWO places. AAA: same angles, a different size.

angle A35°side AC7

SAS: proves congruence. Two sides AND the angle BETWEEN them: the third side is forced. Congruent. The given parts are in red/shaded.

Rule · Congruent = an exact copy

Every side and every angle matches its partner. The ORDER of the letters is the matching: in △ABC ≅ △DEF, A↔D, B↔E, C↔F, and AB = DE, BC = EF, CA = FD.

🧑‍🏫 Rule 4 · Matching partsstep 1 of 5

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📋 The question · SAT · June 2023 · Math 800 Book Q1
Triangles $A B C$ and $D E F$ are congruent, where $A$ corresponds to $D$ , and $B$ and $E$ are right angles. The measure of angle $A$ is $62°$ . What is the measure of angle $F$ ?
1. A) $28°$
2. B) $62°$
3. C) $90°$
4. D) $118°$
follow the letters
$F \leftrightarrow C$
June 2023: $C$ is the third letter of $A B C$ , $F$ the third of $D E F$ .

🧑‍🏫 Rule 5 · Which facts prove congruencestep 1 of 12

1
the four that work
$SSS, SAS, ASA, AAS$
In SAS the angle must be BETWEEN the two sides; in ASA the side is between the two angles.

🎛 Move it · the parallel cut and the bowtie

Slide the cut. The small triangle keeps the angles; every side shrinks by k; the area by k². Then switch to the bowtie.

cut at k = AD/AB0.5

DE ∥ BC makes the corner angles equal (corresponding angles) and ∠A is shared → △ADE ~ △ABC by AA.

Every side × k: AD/AB = AE/AC = DE/BC = 0.5. Perimeter × k too. But area × k² = 0.25 — the small triangle is 25% of the big one, not 50%.

🧑‍🏫 Rule 6 · AA: two equal angles are enoughstep 1 of 9

1
why
$two angles equal \Rightarrow the third too$
Same three angles = same shape. Two angles prove similarity.

🧑‍🏫 Rule 7 · Sides × k, perimeter × k, area × k², angles unchangedstep 1 of 7

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📋 The question · SAT · August 2024 · Math 800 Book Q25
Triangle $QR S$ is similar to triangle $F G H$ such that $Q$ , $R$ , and $S$ correspond to $F$ , $G$ , and $H$ , respectively. Each side of triangle $F G H$ has $\frac{1}{4}$ the length of its corresponding side in triangle $QR S$ . In triangle $F G H$ , the measure of angle $F$ is $35°$ , the measure of angle $G$ is $8°$ , and $F G = a$ . In triangle $QR S$ , the measure of angle $Q$ is $b °$ and $QR = 20$ . What is the value of $a + b$ ?
(type the answer)
sides
$a = \frac{1}{4} \cdot 20 = 5$
August 2024.

🧑‍🏫 Rule 8 · The bowtie (an X between parallels)step 1 of 6

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📋 The question · SAT · November US 2024 · Math 800 Book Q39
$figure from the test$
In the figure, $\overline{L Q}$ intersects $\overline{M P}$ at point $R$ , and $\overline{L M}$ is parallel to $\overline{P Q}$ . The lengths of $M R$ , $L R$ , and $R P$ are 9, 10, and 19, respectively. What is the length of $L Q$ ?
1. A) $\frac{280}{19}$
2. B) $\frac{190}{9}$
3. C) $\frac{271}{9}$
4. D) $\frac{280}{9}$
match across R
$△ L M R \sim △ QP R$
November US 2024: alternate angles + vertical angles.

🎛 Move it · dilation

Scale the shape: sides × k, angles fixed.

scale factor k1.6×

Similar figures: every SIDE is multiplied by k = 1.6, but every ANGLE stays exactly the same. So when PQRS ~ WXYZ, ∠S = ∠Z whatever the side lengths say — the lengths (25 and 5, 45 and 9) are there to distract.

🧑‍🏫 Rule 9 · Dilationstep 1 of 3

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📋 The question · SAT · March US 2026 (A) · Q16
In triangle $R S T$ , the length of $R S$ is 21, and the length of $S T$ is 9. Triangle $R S T$ is dilated by a scale factor of $\frac{1}{3}$ to obtain triangle $R^{'} S^{'} T^{'}$ . What is the length of $S^{'} T^{'}$ ?
1. A)3
2. B)7
3. C)27
4. D)63
every side × k
$S^{'} T^{'} = \frac{1}{3} \cdot S T = 3$
March US 2026 (A): use $S T$ , not $R S$ . A.

🎛 Move it · the altitude to the hypotenuse

Change the legs. The blue, yellow and whole triangles stay similar.

leg AB6leg BC8

The altitude BD splits the right triangle into two smaller right triangles, and all three are similar (each shares an acute angle with the big one).

Big vs yellow (both hold ∠C): BC/BD = AC/AB → 1.67 = 1.67. So BD = AB·BC/AC = 4.8 — legs multiplied, over the hypotenuse (area two ways).

Also BD² = AD·DC: 23.04 = 3.6 × 6.4.

🧑‍🏫 Rule 10 · Three similar right trianglesstep 1 of 4

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📋 The question · SAT · November 2024 · Math 800 Book Q36
$figure from the test$
For triangle $A B C$ , the length of $A B$ is 11 less than the length of $A C$ . Point $D$ (not shown) lies on $\overline{A C}$ such that $\overline{B D}$ (not shown) is perpendicular to $\overline{A C}$ . What is the value of $\frac{B C}{B D}$ ?
(type the answer)
the sides
$A B = 32 - 11 = 21$
November 2024.

🧑‍🏫 Rule 11 · Similarity with real numbersstep 1 of 6

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📋 The question · SAT · October US 2024 · Math 800 Book Q35
$figure from the test$
In the figure shown $A B = 89$ units, $A C = 8$ units, and $C E = 24$ units. What is the area, in square units, of triangle $A D E$ ?
(type the answer)
Pythagoras first
$B C = 89 - 64 = 5$
October US 2024.

🎛 Move it · squares on the sides

Change the legs. Then open or close angle C — c² grows past or falls below a² + b².

leg a3leg b4angle C90°

a² + b² = 25, and c² = 25 → equal: a RIGHT triangle (a² + b² = c²).

Whole-number triple! 3-4-5. Learn the families: 3-4-5, 5-12-13, 8-15-17, 7-24-25 — and any multiple (6-8-10, 10-24-26).

The hypotenuse is always the LONGEST side — a leg can never be longer than it.

🧑‍🏫 Rule 12 · a² + b² = c²step 1 of 6

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📋 The question · SAT · November 2024 · Math 800 Book Q37
In triangle $A B C$ , angle $B$ is a right angle. The length of side $A B$ is $1039$ and the length of side $B C$ is $2439$ . What is the length of side $A C$ ?
1. A) $1439$
2. B) $2639$
3. C) $3439$
4. D) $24 \cdot 39$
take out the common factor
$10, 24, 26 = 2 (5, 12, 13)$
November 2024: $2639$ . B.

🧑‍🏫 Rule 13 · The conversestep 1 of 1

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📋 The question · Gen Z Study · converse of Pythagoras
A triangle has sides 9, 40, and 41. Which statement is true?
1. A)It is a right triangle.
2. B)It is acute.
3. C)It is obtuse.
4. D)No triangle has these sides.
test
$9^{2} + 4 0^{2} = 1681 = 4 1^{2}$
It works backwards: if $a^{2} + b^{2} = c^{2}$ , the triangle IS right. Smaller $c^{2}$ : acute; bigger: obtuse.

🧑‍🏫 Rule 14 · A leg on the xy-planestep 1 of 2

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📋 The question · SAT · November US 2024 · Math 800 Book Q43
$figure from the test$
The line segment shown in the $x y$ -plane represents one of the legs of a right triangle. The area of this triangle is $4813$ square units. What is the length, in units, of the other leg of this triangle?
1. A)16
2. B)32
3. C) $313$
4. D)24
distance
$(4 - (- 5))^{2} + (9 - 3)^{2} = 117 = 313$
November US 2024: the segment from $(- 5, 3)$ to $(4, 9)$ .

🎛 Move it · the special triangles

Switch between the three shapes and change the side. The ratios never change.

leg x4

45-45-90 = half a square. Legs equal (x), hypotenuse x√2 — the square’s diagonal.

Perimeter = 2x + x√2 = 13.66. Seeing “18 + 18√2”? Factor: x(2 + √2) = x√2(√2 + 1), so the HYPOTENUSE x√2 = 18. Going back: leg = hypotenuse ÷ √2.

🧑‍🏫 Rule 15 · 45-45-90: x, x, x√2step 1 of 6

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📋 The question · SAT · October US 2024 · Math 800 Book Q34
$figure from the test$
In the right triangle shown, what is the value of $x$ ?
1. A) $2$
2. B) $37$
3. C) $74$
4. D) $372$
equal legs
$x = 372$
October US 2024: D.

🧑‍🏫 Rule 16 · 30-60-90: x, x√3, 2xstep 1 of 2

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📋 The question · Gen Z Study · 30-60-90
In a 30-60-90 triangle the hypotenuse is 14. What is the length of the longer leg?
1. A)7
2. B) $72$
3. C) $73$
4. D) $143$
from the hypotenuse
$x = 7, x 3 = 73$
Halve, then $\times 3$ : C.

🧑‍🏫 Rule 17 · The equilateral trianglestep 1 of 11

1
height
$h = \frac{3}{2} s$
Half of it is a 30-60-90 with short leg $\frac{s}{2}$ .

🎛 Move it · area

Slide the top: the area holds. Then try two sides and the angle between them.

slide the top160base10height6

Area = ½ · base · height = ½ · 10 · 6 = 30. Slide the top: the triangle leans, the area does NOT change. The height is the PERPENDICULAR distance — it can even fall outside the triangle (an obtuse triangle). In a right triangle the two legs ARE a base and its height.

🧑‍🏫 Rule 18 · ½ · base · heightstep 1 of 4

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📋 The question · SAT · November 2023 · Math 800 Book Q7
A triangle has a base length of 82 centimeters and a height of 164 centimeters. What is the area, in square centimeters, of the triangle?
1. A)246
2. B)3,362
3. C)6,724
4. D)13,448
straight in
$\frac{1}{2} \cdot 82 \cdot 164 = 6724$
November 2023: C. (D forgot the half.)

🧑‍🏫 Rule 19 · Area with algebrastep 1 of 5

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📋 The question · SAT · August US 2023 · Math 800 Book Q6
The area of a triangle is equal to $x^{2}$ square inches. The base of the triangle is $5 + 2 x$ inches, and the height of the triangle is $x - 2$ inches. What is the value of $x$ ?
1. A)2.5
2. B)2.7
3. C)5
4. D)10
set up
$\frac{1}{2} (5 + 2 x) (x - 2) = x^{2}$
August US 2023: double both sides.

🧑‍🏫 Rule 20 · ½ · a · b · sin Cstep 1 of 2

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📋 The question · SAT · December 2024 · Math 800 Book Q48
$figure from the test$
In the figure shown, the measure of angle $X$ is $52°$ . The length of $\overline{X Y}$ is 24 units and the length of $\overline{X Z}$ is 17 units. What is the area, in square units, of triangle $X Y Z$ ?
1. A)204
2. B)408
3. C) $204 sin 52°$
4. D) $408 sin 52°$
the height
$h = 17 sin 52°$
December 2024: drop the height from $Z$ — it is opposite the 52° angle.

🎛 Move it · SOH · CAH · TOA

Pick a ratio. Grow the triangle — the ratio holds. Change the angle — it changes.

angle A37°size (scale)1×

sin A = opposite / hypotenuse = 6.02 / 10 = 0.6 (SOH · CAH · TOA).

Drag the size: every side grows, the ratio does NOT move — it belongs to the ANGLE. That is why similar triangles have equal sin, cos, tan. Drag the angle: now it changes.

Given a ratio like cos A = 3/5? Build the sides as 3 parts and 5 parts (and 4 by Pythagoras), then find what one part is worth.

🧑‍🏫 Rule 21 · A ratio gives the sides in partsstep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q14
In $△ A B C$ , $∠ B$ is a right angle and the length of $\overline{B C}$ is 248 millimeters. If $cos A = \frac{3}{5}$ , what is the length, in millimeters, of $\overline{A B}$ ?
1. A)62
2. B)186
3. C)248
4. D)310
build it
$cos A = \frac{3}{5} : A B = 3 p, A C = 5 p, B C = 4 p$
March 2024: adjacent 3 parts, hypotenuse 5, so the opposite side is 4 (3-4-5).

🧑‍🏫 Rule 22 · Similar triangles have the same ratiosstep 1 of 3

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📋 The question · SAT · June 2023 · Math 800 Book Q2
Triangle $A B C$ is similar to triangle $D E F$ , where angle $A$ corresponds to angle $D$ and angles $C$ and $F$ are right angles. The length $A B$ is 2.4 times the length of $D E$ . If $tan A = \frac{21}{20}$ , what is the value of $sin D$ ?
(type the answer)
same angle, same ratio
$sin D = sin A$
June 2023: the 2.4 is a distractor.

🎛 Move it · two angles of a right triangle

The red side is opposite A and next to B.

angle A35°

The red side is opposite A and next to B. So sin A (opposite/hypotenuse) and cos B (adjacent/hypotenuse) are the SAME fraction. That is why sin A = cos B exactly when A + B = 90° — the two acute angles of one right triangle. (“co-sine” = the sine of the COmplement.)

🧑‍🏫 Rule 23 · sin A = cos B when A + B = 90°step 1 of 3

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📋 The question · SAT · March 2024 · Math 800 Book Q13
In right triangle $R S T$ , the sum of the measures of angle $R$ and angle $S$ is 90 degrees. The value of $sin (R)$ is $\frac{4 2}{9}$ . What is the value of $cos (S)$ ?
1. A) $\frac{4 2}{9}$
2. B) $\frac{4 2}{7}$
3. C) $\frac{7 2}{8}$
4. D) $\frac{9 2}{8}$
complementary
$cos S = sin R = \frac{4 2}{9}$
March 2024: A — no computing.

Real question 1SAT · December 2024 · Math 800 Book Q49type the answer

$figure from the test$

A property owner wants to find the length $x$ , in feet, across a pond as represented in the figure. $P R$ intersects $S T$ at point $Q$ , and $∠ P T Q$ is congruent to $∠ R S Q$ . The lengths represented by $P Q$ , $T Q$ , $QS$ , and $R S$ were determined to be 4,200 feet, 8,400 feet, 1,400 feet, and 1,600 feet, respectively. What is the value of $x$ ?

Take with you

▸Angles add to 180°; isosceles: equal sides face equal angles; exterior angle = the two far angles; |a − b| < third side < a + b; longest side faces the largest angle.
▸Congruent: SSS, SAS, ASA, AAS (the angle BETWEEN, the side matched in both). SSA and AAA do not prove it. Follow the letter order for matching parts.
▸Similar: two equal angles (AA). Parallel cut and bowtie. Sides and perimeter × k, area × k², angles unchanged. Dilation is the same idea.
▸Altitude to the hypotenuse: three similar right triangles — hypotenuse/leg ratios match.
▸a² + b² = c² (and backwards). Triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples. A diameter makes a right angle.
▸45-45-90: x, x, x√2. 30-60-90: x, x√3, 2x. Equilateral: h = (√3/2)s, area (√3/4)s², circumradius s/√3.
▸Area ½·b·h (the height is perpendicular), or ½·a·b·sin C. SOH-CAH-TOA; a ratio builds the sides in parts; sin A = cos B when A + B = 90.

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