Here are the step-by-step solutions for each problem on the worksheet. The main rule we use for all of these is the
Triangle Sum Theorem, which says that the three angles inside any triangle always add up to
180°.
Problem 1
*
Identify the clues: The tick marks on two sides mean this is an
isosceles triangle. This means the two base angles are equal. Since one base angle is $x^\circ$, the other bottom angle is also $x^\circ$. The top angle is $62^\circ$.
*
Set up the equation: $x + x + 62 = 180$
*
Solve:
$$2x + 62 = 180$$
$$2x = 118$$
$$x = 59$$
*
Missing Angles: The two missing angles are both
59°.
Problem 2
*
Identify the clues: The angles are $80^\circ$, $(2x)^\circ$, and $x^\circ$.
*
Set up the equation: $80 + 2x + x = 180$
*
Solve:
$$80 + 3x = 180$$
$$3x = 100$$
$$x = 33.33$$ (or $33 \frac{1}{3}$)
*
Missing Angles:
Angle $x$ is
33.3°.
Angle $2x$ is $2 \times 33.33 =$
66.7°.
Problem 3
*
Identify the clues: The square symbol in the corner means it is a
right angle ($90^\circ$). The other angles are $49^\circ$ and $x^\circ$.
*
Set up the equation: $90 + 49 + x = 180$
*
Solve:
$$139 + x = 180$$
$$x = 41$$
*
Missing Angle: $x$ is
41°.
Problem 4
*
Identify the clues: The angles are $36^\circ$, $61^\circ$, and $(x + 34)^\circ$.
*
Set up the equation: $36 + 61 + (x + 34) = 180$
*
Solve:
Combine the numbers: $36 + 61 + 34 = 131$
$$131 + x = 180$$
$$x = 49$$
*
Missing Angle: Plug $x$ back into the expression $(x + 34)$:
$49 + 34 =$
83°.
Problem 5
*
Identify the clues: The angles are $27^\circ$, $(x + 15)^\circ$, and $x^\circ$.
*
Set up the equation: $27 + (x + 15) + x = 180$
*
Solve:
Combine like terms ($x + x$ and $27 + 15$):
$$2x + 42 = 180$$
$$2x = 138$$
$$x = 69$$
*
Missing Angles:
Angle $x$ is
69°.
Angle $(x + 15)$ is $69 + 15 =$
84°.
Problem 6
*
Identify the clues: The angles are $(2x - 20)^\circ$, $(x + 40)^\circ$, and $(2x + 20)^\circ$.
*
Set up the equation: $(2x - 20) + (x + 40) + (2x + 20) = 180$
*
Solve:
Combine the $x$'s: $2x + x + 2x = 5x$
Combine the numbers: $-20 + 40 + 20 = 40$
$$5x + 40 = 180$$
$$5x = 140$$
$$x = 28$$
*
Missing Angles:
Angle $(2x - 20)$: $2(28) - 20 = 56 - 20 =$
36°.
Angle $(x + 40)$: $28 + 40 =$
68°.
Angle $(2x + 20)$: $2(28) + 20 = 56 + 20 =$
76°.
Problem 7
*
Identify the clues: We know $\angle ABC = 107^\circ$ and $\angle BCA = 36^\circ$. We need to find $\angle CAB$.
*
Set up the equation: $107 + 36 + \angle CAB = 180$
*
Solve:
$$143 + \angle CAB = 180$$
$$\angle CAB = 37$$
*
Answer: The measure of $\angle CAB$ is
37°.
Problem 8
*
Identify the clues: The angles are $(x + 52)^\circ$, $(x + 8)^\circ$, and $x^\circ$.
*
Set up the equation: $(x + 52) + (x + 8) + x = 180$
*
Solve:
Combine $x$'s: $x + x + x = 3x$
Combine numbers: $52 + 8 = 60$
$$3x + 60 = 180$$
$$3x = 120$$
$$x = 40$$
*
Find all angles:
$\angle FDE$ ($x$) =
40°
$\angle DEF$ ($x + 52$) = $40 + 52 =$
92°
$\angle EFD$ ($x + 8$) = $40 + 8 =$
48°
──────────────────────────────────────
Final Answer:
1. $x = 59$; Missing angles are
59° and
59°.
2. $x = 33.3$; Missing angles are
33.3° and
66.7°.
3. $x = 41$; Missing angle is
41°.
4. $x = 49$; Missing angle is
83°.
5. $x = 69$; Missing angles are
69° and
84°.
6. $x = 28$; Missing angles are
36°,
68°, and
76°.
7. $m\angle CAB =$
37°.
8. $x = 40$; Angles are
40°,
92°, and
48°.
Parent Tip: Review the logic above to help your child master the concept of triangle sum theorem worksheet.
![Printable Triangle Sum Theorem Worksheets [PDFs] Brighterly](/images2/triangle-sum-theorem-worksheet/triangle-sum-theorem-worksheet-6.jpg)
