Angle Relationships

📐 Angle Relationships

Part 1 of 5 — Angle Basics & Adjacent Angles

Topics in This Part

🔑 Key Concept: Almost every Grade 7 angle problem comes down to one big idea — angles that fit together add up to a known total, usually $90^\circ$, $180^\circ$, or $360^\circ$. Part 1 builds the vocabulary so the rest of the lesson is careful arithmetic and a little algebra.

What Is an Angle?

An angle is the "opening" between two rays that share a common endpoint. That shared endpoint is the vertex, and the two rays are the sides.

We measure an angle in degrees, written with the little circle symbol: $^\circ$.

• A full turn all the way around is $360^\circ$.
• A half turn (a straight line) is $180^\circ$.
• A quarter turn (a square corner) is $90^\circ$.

We often name an angle with three points, like $\angle ABC$, where the middle letter is always the vertex. So in $\angle ABC$, the vertex is $B$.

💡 Reading tip: The little square drawn in a corner means exactly $90^\circ$ — a right angle. No square means you cannot assume it is $90^\circ$.

Classifying Angles by Size

Examples: $35^\circ$ is acute, $90^\circ$ is right, $128^\circ$ is obtuse, and $180^\circ$ is straight.

🔑 Memory hook: "Acute" angles are "a cute" little angles — small and under $90^\circ$. Obtuse angles are the big, wide ones over $90^\circ$.

Concept Check 🎯

Adjacent Angles

Two angles are adjacent when they:

1. share the same vertex, and
2. share a common side (a ray between them), and
3. do not overlap (one is not inside the other).

Think of two slices of pizza sitting side by side — they touch along one straight cut but do not overlap.

The big idea you will use all lesson long:

🔑 Adding Adjacent Angles: When two angles sit next to each other with no gap and no overlap, their measures add together to give the whole angle.

$\angle ABD + \angle DBC = \angle ABC$

Example: If two adjacent angles measure $40^\circ$ and $25^\circ$, the whole angle they form is $40^\circ + 25^\circ = 65^\circ$.

Adjacent or Not? 🔽

Adjacent angles must share a vertex and a common side and not overlap. Choose whether each described pair is adjacent.

From Adjacent Angles to Totals

Once you can add adjacent angles, the next step is recognizing what total they should reach. A few totals come up again and again:

Keep these three totals in mind — they are the keys to the rest of the lesson. Let's practice adding adjacent pieces.

Add the Adjacent Angles 🧮

Two adjacent angles together form one larger angle. Find each missing measure (in degrees — just type the number).

1) Adjacent angles of $30^\circ$ and $45^\circ$ form a whole angle of $\,?$ 2) A whole angle of $80^\circ$ is split into a $50^\circ$ piece and another piece of $\,?$ 3) Adjacent angles of $18^\circ$ and $72^\circ$ form a whole angle of $\,?$

Wrapping Up Part 1

You now know how to name, classify, and add angles. That last skill — adjacent angles adding up — is the engine for everything ahead.

In Part 2 we meet complementary and supplementary angles: special pairs that add to $90^\circ$ and $180^\circ$. Once you spot those totals, finding a missing angle is just subtraction.

📐 Angle Relationships

Part 2 of 5 — Complementary & Supplementary Angles

🔑 The Idea: Two special pairs show up everywhere. Complementary angles add to $90^\circ$. Supplementary angles add to $180^\circ$. If you know one angle in the pair, you can always find the other by subtracting.

The Two Big Pairs

Complementary ($90^\circ$)

If $\angle A$ and $\angle B$ are complementary, then $\angle A + \angle B = 90^\circ$.

Example: The complement of $35^\circ$ is $90^\circ - 35^\circ = 55^\circ$.

Supplementary ($180^\circ$)

If $\angle A$ and $\angle B$ are supplementary, then $\angle A + \angle B = 180^\circ$.

Example: The supplement of $110^\circ$ is $180^\circ - 110^\circ = 70^\circ$.

💡 Spelling memory: "Complementary" comes before "Supplementary" in the alphabet, just like $90$ comes before $180$ on the number line. C = 90, S = 180.

Concept Check 🎯

Name That Pair 🔽

For each pair of angle measures, choose whether they are complementary, supplementary, or neither.

Finding a Missing Partner

Naming the pair is half the job. The other half is finding the missing angle, and it is always one subtraction:

$\text{missing complement} = 90^\circ - \text{known angle}$ $\text{missing supplement} = 180^\circ - \text{known angle}$

Example: If two angles are supplementary and one is $58^\circ$, the other is $180^\circ - 58^\circ = 122^\circ$.

💡 Decide the total first ($90^\circ$ or $180^\circ$), then subtract. Try a few below.

Find the Missing Angle 🧮

Use subtraction to find each missing partner (type the number of degrees).

1) The complement of $25^\circ$ is $\,?$ 2) The supplement of $90^\circ$ is $\,?$ 3) The complement of $1^\circ$ is $\,?$ 4) The supplement of $173^\circ$ is $\,?$

Wrapping Up Part 2

⚠️ Common mix-up: Do not confuse complementary with supplementary. A right angle ($90^\circ$) is the total for complementary; a straight line ($180^\circ$) is the total for supplementary.

So far the missing angle has always been a plain number. In Part 3 we meet vertical angles and then start using variables like $x$ — that is where the algebra begins.

📐 Angle Relationships

Part 3 of 5 — Vertical Angles & Angles on a Line

🔑 Two new facts: (1) When two straight lines cross, the angles directly across from each other are equal — these are vertical angles. (2) Angles in a row that form a straight line add up to $180^\circ$ — a linear pair.

Vertical Angles

When two straight lines cross, they make an "X" shape with four angles. The angles that sit directly across from each other (not next to each other) are called vertical angles, and they are always equal.

$\text{Vertical angles are congruent (equal): } \angle 1 = \angle 3 \text{ and } \angle 2 = \angle 4$

The angles that sit next to each other along a straight line form a linear pair, and they are supplementary (add to $180^\circ$).

Example

Two lines cross. One of the four angles is $70^\circ$.

• The angle directly across is also $70^\circ$ (vertical angles).
• Each angle next to the $70^\circ$ is $180^\circ - 70^\circ = 110^\circ$ (linear pair).
• And the fourth angle, across from a $110^\circ$, is also $110^\circ$.

💡 The four angles always come in two matching pairs, and the four of them go all the way around: $70 + 110 + 70 + 110 = 360^\circ$. ✓

Concept Check 🎯

Angles on a Straight Line 🔽

Three angles sit in a row along a straight line, so all three add to $180^\circ$. Two of them are $50^\circ$ and $90^\circ$.

All Four Angles at a Crossing

When two lines cross, knowing one angle gives you all four. Here is the routine:

1. The angle directly across is equal (vertical angle).
2. Each angle next to it is $180^\circ - \text{angle}$ (linear pair).
3. The last angle is across from one of those, so it matches it.

The two values always repeat: angle, its supplement, angle, its supplement. Use this routine on the crossing below.

Crossing Lines 🧮

Two straight lines cross, making four angles. One of them is $130^\circ$. Find each of the other three (type the number of degrees).

1) The angle directly across from the $130^\circ$ is $\,?$ 2) An angle next to the $130^\circ$ (linear pair) is $\,?$ 3) The remaining fourth angle is $\,?$

Wrapping Up Part 3

You now have the full toolkit of relationships:

In Part 4 we stop using known numbers for the unknown angle and start calling it $x$. Each rule above becomes a quick equation to solve.

📐 Angle Relationships

Part 4 of 5 — Writing & Solving Equations for Unknown Angles

🔑 The Move: This is the heart of Grade 7 (CCSS 7.G.B.5). Pick the right relationship ($90^\circ$, $180^\circ$, equal, or $360^\circ$), write an equation, and solve for $x$ with the same algebra you already know.

One-Step Equations

Example: Complementary

An angle and its complement: one is $x$, the other is $52^\circ$. They add to $90^\circ$.

$x + 52 = 90 \;\Rightarrow\; x = 90 - 52 = 38^\circ$

Example: Supplementary

Angles $x$ and $135^\circ$ form a straight line, so they add to $180^\circ$.

$x + 135 = 180 \;\Rightarrow\; x = 180 - 135 = 45^\circ$

Example: Vertical

Two vertical angles are $x$ and $77^\circ$. Vertical angles are equal, so:

$x = 77^\circ$

💡 Strategy: First decide which total fits ($90$, $180$, equal, or $360$). Then the equation almost writes itself.

Solve for $x$ 🧮

Write the right equation in your head, then solve (type the number of degrees).

1) $x$ and $66^\circ$ are complementary: $x = \,?$ 2) $x$ and $48^\circ$ are supplementary: $x = \,?$ 3) $x$ and $103^\circ$ are vertical angles: $x = \,?$

Two-Step Equations

Sometimes both angles contain $x$, or there are extra numbers. Add up the parts, set them equal to the right total, then solve.

Example: Two angles that make a right angle

Two complementary angles measure $2x$ and $x$. Together they make $90^\circ$.

$2x + x = 90 \;\Rightarrow\; 3x = 90 \;\Rightarrow\; x = 30$

So the angles are $2x = 60^\circ$ and $x = 30^\circ$. (Check: $60 + 30 = 90$. ✓)

Example: Supplementary with a constant

Two supplementary angles are $(x + 20)^\circ$ and $x^\circ$.

$(x + 20) + x = 180 \;\Rightarrow\; 2x + 20 = 180 \;\Rightarrow\; 2x = 160 \;\Rightarrow\; x = 80$

So the angles are $100^\circ$ and $80^\circ$. (Check: $100 + 80 = 180$. ✓)

⚠️ Watch out: $x$ is usually not the final answer by itself. Plug $x$ back in to report the actual angle measures, and always check that your two measures add to the right total.

Concept Check 🎯

Two-Step Solving 🧮

Set up the equation, solve for $x$, then find the angles where asked.

1) Complementary angles $4x$ and $x$: solve $4x + x = 90$. $x = \,?$ 2) Supplementary angles $(x+30)$ and $x$: solve $(x+30)+x = 180$. $x = \,?$ 3) For problem 2, the larger angle measures $\,?$ degrees.

📐 Angle Relationships

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) classify angles, (2) use complementary and supplementary pairs, (3) use vertical angles and linear pairs, and (4) write and solve equations for unknown angles. Let's put it all together.

Quick Reference

⚠️ Two reminders: (1) Vertical angles are equal, not supplementary — do not subtract from $180^\circ$. (2) After solving for $x$, plug it back in and check that your angle measures add to the correct total.

Pick the Right Total 🔽

For each situation, choose the total the angles must add to (or "equal" if they are vertical angles).

Your Three-Step Strategy

For any unknown-angle problem, run this routine:

1. Spot the relationship — complementary, supplementary, vertical, or around a point?
2. Write the equation — set the parts equal to $90$, $180$, $360$, or to each other.
3. Solve and check — find $x$, plug it back in, and confirm the totals work.

Apply all three steps to the mixed problems below.

Mixed Practice 🧮

Find each unknown (type the number of degrees).

1) $x$ and $34^\circ$ are complementary: $x = \,?$ 2) Two vertical angles are $5x$ and $80^\circ$: solve $5x = 80$, so $x = \,?$ 3) Supplementary angles $(2x)$ and $(x)$: solve $2x + x = 180$, so $x = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.