Geometry Workbook — Chapter 4: Sides and Angles of a Triangle

SOL G.TR.1 (a, b, c, d, e) · Companion to Textbook Chapter 4

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 108.

PAGE 1 — Chapter opener

Chapter 4 · Sides and Angles of a Triangle

Standard G.TR.1 (a, b, c, d, e)

In this chapter you will:

Use the Angle Sum and Exterior Angle theorems to find a missing angle
Order the sides when you know the angles
Order the angles when you know the sides
Decide whether three lengths can form a triangle
Give the range the third side must lie in
Choose the right relationship for a problem in context

Words to know: interior angle · exterior angle · remote interior angles · opposite side · ordering · Triangle Inequality · degenerate · third-side range · isosceles · equilateral · equiangular

Convention: in $\triangle ABC$ , side $a$ is $\overline{BC}$ , side $b$ is $\overline{AC}$ , side $c$ is $\overline{AB}$ — each lowercase letter names the side opposite its vertex. Write ties with an equals sign.

PAGE 2 — The angle sum

4.1 The Angle Sum

FIGURE: fig1-triangle-angle-sum-proof.png (full width)

Fill in the blanks.

The three interior angles of a triangle sum to ____________ .

A triangle has at most ______ angle of $90°$ or more.

The three measures: ______ , ______ , ______ · Sum: ______
Which Chapter 2 theorem justifies the two angles reappearing at $C$ ?
Why must the auxiliary line be parallel to $\overline{AB}$ ?

PAGE 3 — The proof, written out

Why It Is True

Complete the proof that $m\angle A + m\angle B + m\angle C = 180°$ .

Statements	Reasons
1. Draw the line through $C$ parallel to $\overline{AB}$	1. ______________________
2. $\angle A \cong$ angle at $C$ (left)	2. ______________________
3. $\angle B \cong$ angle at $C$ (right)	3. ______________________
4. The three angles at $C$ form a straight angle	4. ______________________
5. $m\angle A + m\angle C + m\angle B = 180°$	5. ______________________

PAGE 4 — The exterior angle

Exterior Angles

FIGURE: fig2-exterior-angle-theorem.png (full width)

Fill in the blanks.

An ____________________ angle equals the sum of its two ____________________ angles.

The exterior angle and its adjacent interior angle form a ____________________ , so they are ____________________ .

Exterior angle measure: ______ Remote interior angles: ______ and ______
Verify by adding: ______ + ______ = ______
State the Exterior Angle Theorem in one sentence.

PAGE 5 — Practice · finding angles

Practice

A-frame: peak ______ · base angles found by ____________________ = ______ each
Third angle: a) $42°$ , $63°$ → ______ b) $90°$ , $37°$ → ______ c) $58°$ , $58°$ → ______
Exterior angle $132°$ → adjacent interior angle ______
Exterior $105°$ , one remote $47°$ → other remote ______
$(x)°$ , $(2x + 10)°$ , $(3x - 4)°$ : $x$ = ______ · angles ______ , ______ , ______
$(5x)°$ , $(6x)°$ , $(7x)°$ : angles ______ , ______ , ______ · classify: ____________
Exterior at $C$ is $(5y + 10)°$ ; remote interiors $(3y)°$ and $(y + 26)°$ . $y$ = ______ · exterior = ______
$71°$ and $46°$ → third angle ______
Exterior $118°$ → adjacent interior ______ · sum of remote interiors ______

PAGE 6 — Apply and reason · Lesson 4.1

Think It Through

FIGURE: fig3-angle-sum-in-context.png (half width, right)

Application. A roof truss has a $52°$ peak and two equal rafters. Each base angle: ______
Application. A garden bed has one angle of $105°$ ; the other two are equal. Each: ______
Error analysis. A student says a triangle can have angles $90°$ , $90°$ , $0°$ . What is wrong?
Error analysis. A student says the exterior angle at $B$ equals $m\angle B$ plus one other angle. Correct it:
Reasoning. Why is an exterior angle always larger than either remote interior angle?
Reasoning. A triangle has a $120°$ angle. What do you know about the other two without calculating?
Can a triangle have angles $88°$ and $94°$ ? ______ Justify: ____________________

PAGE 7 — Longest side, largest angle

4.2 Ordering Sides from Angles

FIGURE: fig4-longest-side-opposite-largest-angle.png (full width)

Fill in the blanks.

The ____________________ side is opposite the ____________________ angle.

Complete the opposite-side frame. In $\triangle ABC$ : side $a$ is ______ , side $b$ is ______ , side $c$ is ______ .

Largest angle: ______ · side opposite it: ______
Smallest angle: ______ · side opposite it: ______
Full side ordering: ____________
State the rule in one sentence: ____________________

PAGE 8 — Find the third angle first

Do Not Skip Step 1

FIGURE: fig5-find-the-missing-angle-first.png (full width)

Fill in the blank. Before ordering, find the ____________________ using the angle sum.

Why can you not order the sides from the left panel alone?
Third angle = ______ · angle ordering: ____________
Side ordering: ____________
Error analysis. Given $m\angle A = 60°$ , $m\angle B = 70°$ , a student writes $a < b$ and says $b$ is longest. Error:

PAGE 9 — Practice · order the sides

Practice

FIGURE: fig12-blank-triangle-frames.png (left third)

Order shortest to longest.

a) $30°$ , $60°$ , $90°$ → ____________

b) $\angle D = 100°$ , $\angle E = 45°$ , $\angle F = 35°$ → ____________

c) $\angle P = 62°$ , $\angle Q = 62°$ , $\angle R = 56°$ → ____________
Find the third angle, then order the sides.

a) $\angle A = 55°$ , $\angle B = 65°$ → third ______ · sides ____________

b) $\angle X = 28°$ , $\angle Y = 34°$ → third ______ · sides ____________

c) $\angle J = 90°$ , $\angle K = 52°$ → third ______ · sides ____________
$\triangle RST$ , $\angle R = \angle S = 47°$ . Longest side as a segment: ______
$\triangle ABC$ , $\angle A = 25°$ , $\angle C = 115°$ . Shortest side as a segment: ______
Angles $89°$ , $46°$ , $45°$ . Longest side: ____________ Right triangle? ______
$\triangle DEF$ : exterior at $F$ is $130°$ , $m\angle D = 62°$ . Side ordering: ____________
$45°$ , $55°$ , $80°$ → ____________
$\angle P = 38°$ , $\angle Q = 52°$ → $m\angle R$ ______ · longest side ______
Side opposite $\angle Y$ in $\triangle XYZ$ : ______

PAGE 10 — Apply and reason · Lesson 4.2

Apply It

Application. A sail has angles $95°$ , $50°$ , $35°$ . Which edge needs the longest rope? ____________
Application. A plot has angles $72°$ , $72°$ , $36°$ . Which side is shortest, and how do you know?
Error analysis. With $m\angle A = 80°$ , a student says $\overline{AB}$ is the longest side. Mistake:
Reasoning. Why must a larger angle face a longer side?
Reasoning. Angles $60°$ , $60°$ , $60°$ — side ordering ____________ · what kind of triangle? ____________

PAGE 11 — Ordering angles from sides

4.3 Ordering Angles from Sides

FIGURE: fig6-ordering-angles-from-sides.png (full width)

Side lengths: $a$ = ______ , $b$ = ______ , $c$ = ______
Sides rank ____________ · angles rank ____________
Largest angle ______ · opposite side ______
State the rule, and say how it relates to Lesson 4.2:

PAGE 12 — Ties

When the Ordering Has a Tie

FIGURE: fig7-isosceles-and-equilateral-ties.png (full width)

Fill in the blanks.

Isosceles: two congruent sides face two congruent angles, so the ordering carries an ____________________ .

Equilateral: all sides equal, so all angles equal ______ .

Which two angles of the isosceles triangle are equal, and why?
Each angle of the equilateral triangle: ______ How do you know? ____________________
Why is " $\angle C < \angle A < \angle B$ " wrong for the isosceles triangle?

PAGE 13 — Practice · order the angles

Practice

a) $a = 4$ , $b = 9$ , $c = 7$ → ____________

b) $a = 15$ , $b = 15$ , $c = 8$ → ____________

c) $a = b = c = 6$ → ____________
$\triangle ABC$ : $AB = 13$ , $BC = 8$ , $AC = 10$ → ____________
$\triangle XYZ$ : $XY = 5$ , $YZ = 12$ , $XZ = 12$ → ____________ (note the tie)
Sides $21$ , $17$ , $29$ — largest angle is opposite ______
$\triangle PQR$ : $PQ = 30$ , $QR = 24$ , $PR = 26$ — smallest angle ______
Sides $9$ , $12$ , $15$ → ordering ____________ · predicted type ____________
$d = e = f = 11$ → ordering ____________ · each angle ______
$a = 7$ , $b = 4$ , $c = 9$ → ____________
Sides $6$ , $6$ , $10$ → ____________
$\overline{AC}$ is longest in $\triangle ABC$ . Largest angle: ______

PAGE 14 — Apply and reason · Lesson 4.3

Apply It

Application. A brace has sides $18$ , $24$ , $30$ in. Widest opening at the corner opposite ______ · side across from it ______
Application. A flowerbed has two sides of $12$ ft and one of $7$ ft. Smallest corner angle is opposite ______
Error analysis. Given $a = 8$ , $b = 5$ , $c = 5$ , a student writes $\angle A < \angle B < \angle C$ . Name two errors:
Reasoning. Why must an equilateral triangle be equiangular?
Reasoning. Given only two side lengths, why can you not order all three angles?

PAGE 15 — Does a triangle exist?

4.4 The Triangle Inequality

FIGURE: fig8-triangle-inequality-test.png (full width)

Fill in the blank. Sort the three lengths and check that the two ____________ ones sum to ____________ than the ____________ .

Left panel lengths: ______ , ______ , ______ · comparison shown: ____________
Why do the two arcs on the right never cross?
State the Triangle Inequality in your own words.
Why is checking only the two shorter lengths enough?

PAGE 16 — The boundary case

When It Flattens

FIGURE: fig9-degenerate-boundary-case.png (full width)

Fill in the blanks.

When the two shorter lengths sum to ____________ the longest, the figure flattens: no ____________ , no ____________ , no ____________ .

That is why the inequality is ____________________ .

$3 + 5 =$ ______ · what does the figure look like? ____________________
Why "greater than" rather than "greater than or equal to"?
Can $5$ , $5$ , $10$ form a triangle? ______ What is special about this case? ____________

PAGE 17 — Practice · does it exist?

Practice

Show the comparison and decide.

a) $5$ , $7$ , $11$ → ____________ ______

b) $8$ , $3$ , $4$ → ____________ ______

c) $10$ , $10$ , $19$ → ____________ ______

d) $6$ , $6$ , $12$ → ____________ ______
a) $2.5$ , $3.5$ , $5$ ______ b) $14$ , $9$ , $22$ ______ c) $1$ , $1$ , $1$ ______
Sides $9$ and $4$ . Which of $4$ , $5$ , $12$ , $13$ works? ______
Sides $11$ and $11$ . Which of $1$ , $21$ , $22$ , $23$ works? ______
Sides $x$ , $8$ , $15$ with $x$ whole. Smallest $x$ = ______
Largest whole $x$ = ______
$6$ , $7$ , $14$ ? ______ Comparison: ____________
$9$ , $12$ , $20$ ? ______
State the single deciding comparison, and why the other two are unnecessary:

PAGE 18 — Apply and reason · Lesson 4.4

Apply It

A student checks $6 + 10 > 8$ and $8 + 10 > 6$ for lengths $6$ , $8$ , $10$ , then stops. Which comparison decides it? ____________ Answer: ______
Application. Boards $18$ , $25$ , $46$ in. Triangular frame? ______ Comparison: ____________
Application. Ropes $12$ , $16$ , $28$ ft. Will it work? ______ If not, how much must the longest be shortened?
Error analysis. A student says $4$ , $9$ , $13$ works because $4 + 13 > 9$ . Error and correct answer:
Reasoning. Why can two sides of length $s$ never form a triangle with a third side of $2s$ or more?
Reasoning. What goes wrong geometrically in the boundary case, and why is "degenerate" a fair name?

PAGE 19 — The range for a third side

4.5 The Third-Side Range

FIGURE: fig10-third-side-range.png (full width)

Fill in the blank. $\lvert p - q \rvert < x <$ ____________ — greater than the ____________ , less than the ____________ . Both endpoints are ____________ .

Two known sides: ______ and ______ · range: ____________
Why are both circles open?
Whole-number values allowed: ____________ Count: ______

FIGURE: fig12-blank-triangle-frames.png (right third — number line)

PAGE 20 — Practice · ranges

Practice

a) $6$ and $10$ → ____________ b) $4$ and $4$ → ____________ c) $15$ and $22$ → ____________
Range and whole-number count.

a) $8$ and $11$ → ____________ · count ______

b) $5$ and $13$ → ____________ · count ______
Sides $9$ and $16$ . Smallest whole third side ______ · largest ______
Sides $x$ , $12$ , $19$ . Range ____________ · whole-number count ______
Error analysis. For sides $7$ and $9$ , a student writes $7 < x < 9$ . Error and correct range:
Reasoning. Why is the lower bound the difference of the two sides?
Sides $11$ and $6$ → range ____________
Whole-number count for item 105: ______

PAGE 21 — Problems in context

Choosing the Right Tool

FIGURE: fig11-triangle-problems-in-context.png (full width)

$m\angle C$ = ______
Shortest side: ____________ Relationship used: ____________________
If $\angle A$ were $70°$ : $m\angle C$ = ______ Would the shortest side change? ______

Complete the tool-choice table.

The problem gives you	Use
Two angles, asks for the third	____________________
An exterior angle and one remote interior	____________________
Angles, asks which side is longest	____________________
Sides, asks which angle is largest	____________________
Three lengths, asks "will it work?"	____________________
Two lengths, asks what the third could be	____________________

Name the relationship for each: (i) ____________ (ii) ____________ (iii) ____________

PAGE 22 — Context practice

Apply It

Application. A sign has angles $48°$ , $48°$ , $84°$ . Longest edge: ____________ Which two are equal? ____________
Application. Struts $14$ in and $9$ in; whole-inch crosspieces. Possible lengths: ____________________
Application. Trail legs $3.5$ km and $5.2$ km. Third leg between ______ and ______ km
Application. $\triangle ABC$ with $m\angle A = 96°$ , $m\angle B = 42°$ . Sides shortest to longest, as segments:
Angles $27°$ , $63°$ , $90°$ → sides shortest to longest: ____________________

PAGE 23 — Chapter review

Review

Review 1 (G.TR.1 a, e). $m\angle A = 41°$ ; exterior angle at $C$ is $118°$ .

$m\angle C$ = ______ (theorem: ____________________ )
$m\angle B$ = ______ (theorem: ____________________ )
Angle ordering: ____________ Side ordering: ____________
Longest side as a segment: ______
Why could you not order the sides without finding a missing angle first?

Review 2 (G.TR.1 b, c, d). Two sides measure $13$ and $6$ .

Range: ____________ Graph it on the number line below, with the correct endpoints.
Whole-number possibilities: ____________________ Count: ______
Which whole number makes the triangle isosceles? ______ Angle ordering then: ____________
For $x = 8$ , angle ordering: ____________

Review 3 (G.TR.1 e). A roof brace with corner angles $53°$ and $89°$ .

Third angle: ______
Sides shortest to longest: ____________________
Boards $4$ , $9$ , $14$ ft — any triangle? ______ Comparison: ____________
Keeping $4$ ft and $9$ ft, range for the third board: ____________ Whole-number options: ____________

Canva production notes

Page size: 8.5 × 11 in, 0.6 in margins. One workbook page per Canva page.
Type: page title H1 28 pt, section label H2 18 pt, body 12 pt, answer blanks 12 pt with a 1 pt rule.
Figures: place at the width noted beside each FIGURE: line. All figures are 200 dpi PNG on white.
Proof frame on page 3 is a typed two-column table with ruled blanks, not a figure.
Blank frames: fig12-blank-triangle-frames.png holds two unlabeled triangles and a $0$ – $20$ number line. Use the triangles on pages 9 and 13 for extra ordering practice and the number line on pages 19 and 23 for graphing a range — open circles at both endpoints.
Symbols: ∠, △, ≅, °, and the absolute-value bars must render in the body font; check the Canva font supports them before the first export.
Item numbers are continuous from 1 to 108 and must not be renumbered when pages are reordered.