Digital SAT · Math · Unit 4 · Geometry & Trigonometry

Lesson 4.1 · Lines & Angles

On the test: 1–2 questions a test — a figure with a few angles, or angles written as expressions; the reference sheet gives you none of these facts

Big idea. Almost every angle question rests on four facts: a straight line is 180°, crossing lines make equal opposite angles, parallel lines cut by a transversal make only TWO sizes of angle, and a triangle’s angles add to 180°. Everything else — polygons, exterior angles, radians — is built from them.

🎛 Move it · two crossing lines

Turn one line. Opposite angles stay equal; neighbours always make 180°.

angle between them50°

Opposite angles are equal (vertical angles): both 50°. Neighbours add to 180° (a straight line): 50 + 130 = 180. All four add to 360°.

Rule · The words

Acute: less than 90° · Right: exactly 90° · Obtuse: between 90° and 180° · Straight: 180°. Complementary angles add to 90°; supplementary angles add to 180°. Perpendicular lines meet at 90°.

🧑‍🏫 Rule 1 · Complement and supplementstep 1 of 3

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📋 The question · Gen Z Study · complementary and supplementary
An angle measures $34°$ . What is the measure of its supplement minus the measure of its complement?
(type the answer)
supplement
$180 - 34 = 146$
The angle that completes a straight line.

🧑‍🏫 Rule 2 · A straight line is 180°; opposite angles are equalstep 1 of 5

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📋 The question · SAT · August US 2024 · Math 800 Book Q20
$figure from the test$
In the figure, line $p$ is parallel to line $q$ , and $t$ intersects both of them. What is the value of $x + 142$ ?
1. A)52
2. B)90
3. C)142
4. D)180
linear pair
$x + 142 = 180$
August US 2024: $x$ and 142° sit side by side on a straight line. D.

🧑‍🏫 Rule 3 · Around a point 360°; a bisector halvesstep 1 of 2

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📋 The question · Gen Z Study · around a point
Four angles around a point measure $70°$ , $95°$ , $110°$ and $x °$ . What is $x$ ?
(type the answer)
all the way round
$70 + 95 + 110 + x = 360$
A full turn: $x = 85$ .

🧑‍🏫 Rule 4 · Crossing-line angles as expressionsstep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q7
Two lines intersect at exactly one point, forming two acute angles and two obtuse angles. The measure of one of these angles is $(9 x - 190) °$ . Which of the following could NOT be the sum of the measures of any two of these angles?
1. A) $(- 18 x + 380) °$
2. B) $(- 18 x + 740) °$
3. C) $(18 x - 380) °$
4. D) $180°$
the two sizes
$9 x - 190 and 180 - (9 x - 190) = 370 - 9 x$
March 2024: the four angles are two of each.

🎛 Move it · parallel lines and a transversal

Click an angle: every equal angle lights up. Click another: the pair is named. Then tilt the lower line and watch the equalities break.

transversal angle55°tilt the lower line0°

Yellow = the angle you picked (top-right at the upper line, 55°). Every angle of the same size is coloured. Click another angle: it becomes yours, and the pair is named below.

With parallel lines there are only two sizes: the acute ones (55°) are all equal, the obtuse ones (125°) are all equal, and an acute + an obtuse = 180°.

🧑‍🏫 Rule 5 · Only two sizes of anglestep 1 of 6

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the big fact
$all acute equal, all obtuse equal, acute + obtuse = 180$
A transversal cuts each parallel line at the same tilt, so the eight angles come in just two sizes.

🧑‍🏫 Rule 6 · The names of the pairsstep 1 of 4

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corresponding — equal
$same corner at each line$
Top-right at one line and top-right at the other.

🧑‍🏫 Rule 7 · Angles written as expressionsstep 1 of 6

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📋 The question · SAT · October US 2024 · Math 800 Book Q23
$figure from the test$
In the figure, line $q$ is parallel to line $r$ , and both lines are intersected by line $s$ . If $y = 2 x + 9$ , what is the value of $x$ ?
1. A)26
2. B)35
3. C)55
4. D)65
decide: same or supplement
$y = 180 - 61 = 119$
October US 2024: $y$ is obtuse, 61° acute — they add to 180.

🧑‍🏫 Rule 8 · Proving two lines parallel; perpendicularstep 1 of 3

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📋 The question · SAT · October US 2024 · Math 800 Book Q28
$figure from the test$
In the figure shown, line $l$ intersects lines $j$ and $k$ . Which additional piece of information is sufficient to prove that lines $j$ and $k$ are parallel?
1. A) $w = 35°$
2. B) $x = 145°$
3. C) $y = 35°$
4. D) $z = 145°$
what proves nothing
$y = 35 always (vertical to the 35°)$
October US 2024: $y$ and $z$ are at the SAME crossing as the 35° — they are fixed whatever $k$ does.

🧑‍🏫 Rule 9 · Three slanted lines: compare their tiltsstep 1 of 5

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📋 The question · SAT · August 2024 · Math 800 Book Q19
$figure from the test$
In the figure, parallel lines $d$ and $e$ are intersected by three other lines. If $d = 42$ , $g = 143$ , and $h > d$ , which statement about $j$ and $p$ must be true?
1. A) $j = p$
2. B) $j + p = 90$
3. C) $j > p$
4. D) $j < p$
each line’s angle with the parallels
$c : 180 - 143 = 37°, right line : 42°, left line : h > 42°$
August 2024: read how steeply each slanted line meets $d$ and $e$ .

🎛 Move it · the triangle

Move the top corner. The three angles change but always add to 180°; the exterior angle always equals the two far angles. Tick isosceles.

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 10 · A triangle’s angles add to 180°step 1 of 2

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📋 The question · SAT · October 2024 · Math 800 Book Q24
In triangle $A B C$ , the measure of angle $A$ is $47°$ , the measure of angle $B$ is $90°$ , and the measure of angle $C$ is $(\frac{k}{2}) °$ . What is the value of $k$ ?
1. A)43
2. B)45
3. C)86
4. D)94
the third angle
$\frac{k}{2} = 180 - 47 - 90 = 43$
October 2024: careful — the question asks for $k$ , not the angle.

🧑‍🏫 Rule 11 · The exterior anglestep 1 of 4

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the fact
$exterior = the two remote interior angles$
Extend one side: the outside angle equals the sum of the two angles NOT next to it.

🧑‍🏫 Rule 12 · Isosceles: equal sides, equal anglesstep 1 of 4

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📋 The question · SAT · October US 2024 · Math 800 Book Q22
In $△ J K L$ , the measures of both $∠ J$ and $∠ K$ are equal, and the measure of $∠ L$ is $134°$ . What is the measure of $∠ J$ ?
1. A) $23°$
2. B) $46°$
3. C) $67°$
4. D) $90°$
the two equal angles
$J = \frac{180 - 134}{2} = 23$
October US 2024: A.

🎛 Move it · the polygon

Change the number of sides: count the triangles from one corner.

number of sides n6

From one corner, the diagonals cut the polygon into 6 − 2 = 4 triangles, each 180°: the inside angles add to (6 − 2)·180 = 720°.

Regular (all angles equal): each = 720/6 = 120° — as a fraction of 180, each is (6 − 2)/6 · 180. The exterior angles always add to 360°: each 60°.

🧑‍🏫 Rule 13 · Polygonsstep 1 of 7

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the sum
$(n - 2) \cdot 180$
Diagonals from one corner make $n - 2$ triangles.

🎛 Move it · degrees and radians

Turn the angle. 180° is π: watch the radian value and the arc.

angle120°radius r3

180° = π radians (half a turn), 360° = 2π. So: degrees → radians: × π/180 · radians → degrees: × 180/π.

A radian is the angle whose arc is exactly one radius long — so the arc is always θ (in radians) × r. Whole turns (2π) can be added to any angle and land in the same place.

🧑‍🏫 Rule 14 · Radians ↔ degreesstep 1 of 6

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the key
$π rad = 180°$
Radians → degrees: $\times \frac{180}{π}$ . Degrees → radians: $\times \frac{π}{180}$ .

🎛 Move it · similar figures

Scale the shape: the sides grow, the angles do not.

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 15 · Similar figures keep their anglesstep 1 of 3

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📋 The question · SAT · November 2023 · Math 800 Book Q5
Quadrilaterals $P QR S$ and $W X Y Z$ are similar, where $P$ , $Q$ , and $R$ correspond to $W$ , $X$ , and $Y$ , respectively. The measure of $∠ S$ is $105°$ , $P S = 25$ , and $W Z = 5$ . What is the measure of $∠ Z$ ?
1. A) $5°$
2. B) $21°$
3. C) $75°$
4. D) $105°$
corresponding angles
$∠ Z = ∠ S = 105°$
November 2023: the side lengths are there to distract. D.

🎛 Move it · two angles of a right triangle

Change angle A. 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 16 · sin A = cos B ⇔ A + B = 90°step 1 of 3

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📋 The question · SAT · August US 2023 · Math 800 Book Q4
For two acute angles, $∠ A$ and $∠ B$ , $cos A = sin B$ . The measures, in degrees, of $∠ A$ and $∠ B$ are $x + 8$ and $4 x + 2$ , respectively. What is the value of $x$ ?
1. A)37
2. B)34
3. C)16
4. D)2
the fact
$cos A = sin B \Rightarrow A + B = 90$
August US 2023: the two acute angles of one right triangle.

Real question 1SAT · June 2024 · Math 800 Book Q14type the answer

$figure from the test$

In the figure, line $n$ intersects lines $r$ and $s$ . Line $r$ is parallel to line $s$ . What is the value of $x$ ?

Take with you

▸Straight line 180°; around a point 360°; opposite (vertical) angles are equal; complementary 90°, supplementary 180°.
▸Parallel lines + a transversal: only two sizes — all acute equal, all obtuse equal, acute + obtuse = 180.
▸Equal corresponding (or alternate) angles PROVE lines are parallel; an angle at the same crossing proves nothing.
▸Triangle: 180°. Exterior angle = the two far interior angles. Isosceles: equal sides face equal angles.
▸Polygon: (n − 2)·180; regular: each (n − 2)·180/n; exterior angles always 360.
▸π rad = 180°: radians → degrees × 180/π. Similar figures keep every angle. sin A = cos B ⇔ A + B = 90.

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