Lines, angles, and triangles

You get a figure, or a sentence describing one, and the last line often reads like one of these:

• “In the figure, line $p$ is parallel to line $q$, and line $t$ intersects both lines. What is the value of $x$?”
• “In the triangle shown, $PQ = QR$. What is the value of $x$?”
• “What is the length of $\overline{XY}$?”, after a sentence saying which corners of two triangles match.
• “Which additional piece of information is sufficient to determine whether triangle $ABC$ is congruentExactly the same shape and size. to triangle $DEF$?”

Figures are drawn to scale unless a note under them says “Figure not drawn to scale.” Nearly every figure in these questions has that note, so go by the numbers and marks only. If a figure has no note, you can estimate from it, so if you get $40^\circ$ for an angle drawn wider than a square corner, recheck your work.

On scratch paper, copy the figure bigger, or draw it if there isn't one, and write every number from the question on it. Then fill in every angle you can get in one step. Desmos won't help here, since there's nothing to graph.

Two crossing lines make four angles. The two across from each other are vertical anglesThe two angles across from each other where two lines cross. They are always equal., and they're always equal. Two angles side by side on one straight line are supplementaryTwo angles that add up to 180 degrees., which means they add to $180^\circ$.

Two angles that share an edge but don't sit on one straight line can add to anything.

A line that crosses two parallel lines is called a transversalA line cutting across two parallel lines.. It cuts both at the same tilt, so the four angles at one crossing are copies of the four at the other.

That leaves only two sizes. If one angle is $75^\circ$, each of the eight is $75^\circ$ or $105^\circ$. Two small angles are equal, two big ones are equal, and a small one plus a big one is $180^\circ$.

Questions may name the pairs, such as corresponding anglesAngles in the matching position where a line crosses two others, such as both upper-right angles. When the two lines are parallel, they are equal..

Some questions run this backward and ask which fact proves two lines parallel. Equal corresponding angles do, and so do equal alternate interior angles, or same-side interior angles that add to $180^\circ$. Next to a $74^\circ$ angle, a $106^\circ$ angle on the same line proves nothing, since that's true either way.

Lines $p$ and $q$ are parallel. The two labeled angles, $(4x-15)^\circ$ and $(2x+9)^\circ$, sit between them, on the same side of $t$, so one is big and one is small. The question asks for $4x-15$.

1. One big and one small angle add to $180^\circ$.

   $$(4x-15)+(2x+9)=180$$

2. Combine $4x+2x$ and $-15+9$.

   $$6x-6=180$$

3. Add 6 to both sides.

   $$6x=186$$

4. Divide both sides by 6.

   $$x=31$$

5. The question asks for $4x-15$, so put in 31.

   $$4x-15=109$$

If a second equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson is printed under the figure, such as $y=3x-25$, and the question asks for $y$, finish the geometry first. Then $y=3(31)-25=68$.

Harder versions ask about a total with no figure. A line crosses two parallel lines, so four angles are acute (under $90^\circ$) and four are obtuse. One acute angle is $(4x-230)^\circ$, and two acute angles plus three obtuse ones add to $(-4x+w)^\circ$. What is $w$?

Try numbers first. If the acute angles were $50^\circ$, the obtuse ones would be $130^\circ$, and two acute plus three obtuse would be $100+390=490$. That's 540 minus the acute angle.

1. Call the acute angle $a$. Each obtuse angle is $180-a$.

   $$2a+3(180-a)$$

2. Multiply out, then $2a-3a=-a$.

   $$540-a$$

3. Put in $a=4x-230$.

   $$540-(4x-230)$$

4. The minus sign flips both signs inside.

   $$540-4x+230$$

5. Add $540+230$.

   $$770-4x$$

6. Match $770-4x$ with $-4x+w$.

   $$w=770$$

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Reyes Nursery uses the alignment shown to position a divider above its compost-mixing bins.

In the figure, line $r$ is parallel to line $s$, and line $t$ intersects both lines. What is the value of $x+34$?

1. A

   84

2. B

   78

3. C

   96

4. D

   50

Show the answer

• A

  84

  Correct answer

  The labeled angles are same-side interior angles, so they are supplementaryTwo angles that add up to 180 degrees.Read the lesson. Solving $(x+34)+(3x-54)=180$ gives $x=50$, and therefore $x+34=84$.

• B

  78

  This results from setting the same-side interior angles equal: $x+34=3x-54$. That equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson gives $x=44$ and $x+34=78$, but these angles are supplementaryTwo angles that add up to 180 degrees.Read the lesson, not equal.

• C

  96

  This results from solving for $x$ correctly and then subtracting the requested expressionNumbers and letters combined with no equals sign, like 5x + 7.Read the lesson from $180$. The value requested is $x+34$, not its supplement $180-(x+34)$.

• D

  50

  This is the value of $x$ alone. The question asks for $x+34$, so you must add $34$ after finding $x=50$.

Solution

The top lower-left angle and the bottom upper-left angle are same-side interior angles. Since $r$ and $s$ are parallel, these angles are supplementaryTwo angles that add up to 180 degrees.Read the lesson:

$$(x+34)+(3x-54)=180$$

Combine like termsParts of an expression with the same letter raised to the same power, like 5x and 2x, which you can add together.Read the lesson and solve:

$$4x-20=180$$ $$4x=200$$ $$x=50$$

The question asks for $x+34$, so substitute $50$:

$$x+34=50+34=84$$

Report a problem

The three angles in any triangle add to $180^\circ$.

In a right triangle, the right angle takes $90^\circ$, so the other two add to $90^\circ$. Two angles that add to $90^\circ$ are complementaryTwo angles that add up to 90 degrees..

An isoscelesA triangle with two equal sides, which also has two equal angles. triangle has two equal sides. The two angles across from those sides, called the base angles, are equal too.

The letters can fool you. With $PQ=QR$, the letter $Q$ is in both names, so $Q$ is the corner between the equal sides, and the equal angles are at $P$ and $R$.

When all three sides are equal, the triangle is equilateral and every angle is $60^\circ$. Three equal angles also mean three equal sides. Go by tick marks or numbers here, since even a figure drawn to scale can't show $60^\circ$ apart from $62^\circ$.

“$\overline{QR}$ is extended through $R$ to point $S$” puts $S$ past $R$, outside the triangle. Draw $Q$, $R$ and $S$ on one line, in that order.

The outside angle $PRS$ and the inside angle at $R$ share a straight line. Take angle $P$ as $50^\circ$ and angle $Q$ as $48^\circ$.

1. The inside angle at $R$.

   $$180-50-48=82$$

2. Angle $PRS$ is beside it on the straight line.

   $$180-82=98$$

3. Or add the inside angles at the other two corners, which is quicker.

   $$50+48=98$$

This one adds equal sides, with no figure.

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At a pencil factory, a guide for graphite cores is set with the alignment diagram shown, in which lines $h$ and $j$ are parallel. The guide also has a triangular outline $RUV$, not shown, with $RU = RV$. The outline’s interior angle at $R$ measures $x^\circ$, where $x$ is the value labeled in the diagram. Side $UV$ extends beyond $V$ to point $W$.

What is the measure of $\angle RVW$, in degrees?

Show the answer

Accepted answers: 142, 142.0

Solution

The two upper-left angles in the alignment diagram are corresponding anglesAngles in the matching position where a line crosses two others, such as both upper-right angles. When the two lines are parallel, they are equal.Read the lesson on parallel lines, so $x = 104$. Since $RU = RV$, the base angles at $U$ and $V$ are equal. Each measures $(180 - 104)/2 = 38^\circ$.

Because $UV$ extends beyond $V$ to $W$, angles $RVU$ and $RVW$ sum to $180^\circ$. Thus, $\angle RVW = 180^\circ - 38^\circ = 142^\circ$.

The wrong answer $104$ comes from subtracting the apex angle from $180^\circ$, treating the remaining $76^\circ$ as one base angle, and taking its supplement. That remaining $76^\circ$ is the sum of both equal base angles.

The wrong answer $76$ comes from treating the $104^\circ$ apex angle as a base angle and taking its supplement. It can also come from taking the supplement of the apex angle when the requested exterior angle is at $V$.

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Similar trianglesTriangles with the same angles, so one is a bigger or smaller copy of the other with every side multiplied by the same number. are the same shape in different sizes. Their matching angles are equal, and each side of one is the same number times the matching side of the other. That number is the scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.. Two pairs of equal angles are enough to make two triangles similar. A triangle “dilated by a scale factor of 3” gives a similar one with every side 3 times as long.

Say triangle $KLM$ is similar to triangle $RST$, with $KL=7$ and $RS=35$, and angle $L$ is $56^\circ$. The scale factor is $35\div7=5$, and a question might write that as $RS=5(KL)$.

The order of the letters tells you which corners match. “Triangle $ABC$ is similar to triangle $XYZ$” means $A$ matches $X$, $B$ matches $Y$, and $C$ matches $Z$. So $AB$ matches $XY$, and $BC$ matches $YZ$. Take $AB=9$, $BC=24$ and $YZ=40$, and find $XY$.

1. Triangle $ABC$'s side on top in both fractions.

   $$\frac{AB}{XY}=\frac{BC}{YZ}$$

2. Fill in the lengths.

   $$\frac{9}{XY}=\frac{24}{40}$$

3. Cross-multiply: $9\times40=360$.

   $$24\cdot XY=360$$

4. Divide both sides by 24.

   $$XY=15$$

If two matching sides add to 100 and the scale factor is 4, the sides are $s$ and $4s$, so $5s=100$, $s=20$, and the long side is 80.

Two crossing segments make a bow tie of two triangles. If $\overline{GK}$ and $\overline{HJ}$ cross at $M$ and angle $G$ equals angle $J$, the vertical angles at $M$ make triangle $GMH$ similar to triangle $JMK$. So $GM$ matches $JM$, on a different segment. Write $GMH$ over $JMK$ before you pair sides.

In triangle $ABC$, $D$ is on side $AB$, $E$ is on side $AC$, and $\overline{DE}$ is parallel to $\overline{BC}$.

The parallel line cuts off a small triangle $ADE$ that's similar to the whole triangle $ABC$. Say $AD=21$, $AB=24$ and $BC=54$. Find $DE$.

1. Small triangle on top, whole triangle below, both sides starting at $A$.

   $$\frac{DE}{BC}=\frac{AD}{AB}$$

2. Fill in the lengths.

   $$\frac{DE}{54}=\frac{21}{24}$$

3. Multiply both sides by 54.

   $$DE=\frac{54\cdot21}{24}$$

4. $54\times21=1{,}134$, and $1{,}134\div24=47.25$.

   $$DE=47.25$$

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Solveig plots a rain gauge’s weekly totals on the triangular panel shown for the station’s field log. $\overline{DE} \parallel \overline{BC}$, and $AB=45$.

What is the length of $\overline{DE}$?

1. A

   8.4

2. B

   13.2

3. C

   33

4. D

   $\frac{110}{3}$

Show the answer

• A

  8.4

  This applies the scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.Read the lesson $\frac{27}{45}$ to the $14^\circ$ angle, giving $8.4$. Similarity preserves angle measures; the scale factor applies to corresponding side lengths.

• B

  13.2

  Correct answer

  Because $\overline{DE} \parallel \overline{BC}$, triangles $ADE$ and $ABC$ are similar. Their scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.Read the lesson is $\frac{AD}{AB}=\frac{27}{45}=\frac{3}{5}$, so $DE=\frac{3}{5}(22)=13.2$.

• C

  33

  This uses $\frac{AD}{DB}=\frac{27}{18}$ as the scale factorThe number every length is multiplied by to turn one figure into a bigger or smaller copy of it.Read the lesson. The similarity ratioA comparison of two amounts, like 3 to 5.Read the lesson must compare corresponding whole sides, so $AD$ corresponds to $AB$, not to the leftover segment $DB$.

• D

  $\frac{110}{3}$

  This inverts the similarity ratioA comparison of two amounts, like 3 to 5.Read the lesson to $\frac{AB}{AD}=\frac{45}{27}$. That produces a value longer than $BC$, even though $DE$ is the corresponding side of the smaller triangle.

Solution

Since $\overline{DE} \parallel \overline{BC}$, corresponding anglesAngles in the matching position where a line crosses two others, such as both upper-right angles. When the two lines are parallel, they are equal.Read the lesson are equal, so $\triangle ADE \sim \triangle ABC$.

The corresponding sides are $AD$ and $AB$, and $DE$ and $BC$. Set up the proportionTwo ratios set equal to each other, which you can solve for a missing number.Read the lesson $$\frac{DE}{BC}=\frac{AD}{AB}.$$ Substitute the given lengths: $$\frac{DE}{22}=\frac{27}{45}=\frac{3}{5}.$$ Therefore, $$DE=22\left(\frac{3}{5}\right)=\frac{66}{5}=13.2.$$

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If the scale factor is $k$, every length grows by $k$, and area grows by $k^2$.

Harder questions make you find a missing side first, often with the Pythagorean theorem, which the right triangles lesson covers. The long side across from a right angle is the hypotenuseThe longest side of a right triangle, the one opposite the right angle.Read the lesson.

In triangle $ABC$, $D$ is on $AB$ and $E$ is on $AC$, with $\overline{DE}$ parallel to $\overline{BC}$. Angles $AED$ and $ACB$ are right angles. $AD=\sqrt{41}$, $AE=4$ and $EC=8$. What is the area of triangle $ABC$?

1. $AE^2+DE^2=AD^2$, since $AD$ is the hypotenuse.

   $$DE^2=41-4^2=25$$

2. Take the square root.

   $$DE=5$$

3. $AC$ is both pieces, 12, so the scale factor is $12\div4$.

   $$k=\frac{4+8}{4}=3$$

4. The other leg grows by $k$ too.

   $$BC=3\cdot5=15$$

5. Area is half of base times height.

   $$\tfrac{1}{2}(12)(15)=90$$

An amount that goes with length, such as perimeterThe distance all the way around a flat shape.Read the lesson or a fence's cost, grows by $k$ once. A $50 fence around a triangle with sides 6, 8 and 11 has a similar twin with long side 44, so $k=4$ and the twin's fence costs $200. Using $k^2$ would give $800.

Congruent triangles are the same shape and size, so every matching side and angle is equal. If triangles $ABC$ and $DEF$ are congruent, with $A$ matching $D$, right angles at $B$ and $E$, and angle $A$ at $44^\circ$, then angle $F$ matches angle $C$, which is $90-44=46^\circ$.

Other questions ask which extra fact is “sufficient to determine” that two triangles are similar or congruent. Sufficient means enough to prove it.

To prove two triangles similar, you need two pairs of equal angles. Find the third angle first. Triangle $ABC$ with $40^\circ$ at $A$ and $75^\circ$ at $B$ has $65^\circ$ left, so it's similar to triangle $DEF$ with $40^\circ$ at $D$ and $65^\circ$ at $E$. Two pairs of sides in the same ratioA comparison of two amounts, like 3 to 5.Read the lesson also work, but only if the angles between those sides are equal. If the equal angles sit anywhere else, the sides prove nothing.

To prove them congruent, the sizes have to match too. Below, S means a pair of equal sides and A a pair of equal angles.

Two sides with an angle that isn't between them (SSA) proves nothing.

Equal angles can't show equal sizes, so which fact would make triangles $ABC$ and $DEF$ above congruent?

With statements I and II, test each one alone with the facts you were given, then both together. “No additional information is needed” is right when those facts already prove it, such as two equal angle pairs on a question about similar triangles.

Before you calculate, say which relationship you're using. When you're done, check your answer against the last line of the question.