Geometry worksheet with problems on the Exterior Angle Theorem and Triangle Sum Theorem, featuring diagrams of triangles with given angles and expressions to solve for unknown values.
Geometry worksheet titled "3.5 Exterior Angle Theorem and Triangle Sum Theorem" with ten problems involving triangles and angles, asking to find the measure of indicated angles and solve for x.
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Step-by-step solution for: Ext Angle THM Practice Triangle Sum Practice PDF | PDF | Geometric ...
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Show Answer Key & Explanations
Step-by-step solution for: Ext Angle THM Practice Triangle Sum Practice PDF | PDF | Geometric ...
Here are the step-by-step solutions for each problem on the worksheet.
1)
* Concept: The Exterior Angle Theorem states that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
* Calculation: The exterior angle at $C$ (angle $GCD$) is equal to angle $D$ plus angle $E$.
$$? = 70^\circ + 24^\circ$$
$$? = 94^\circ$$
2)
* Concept: Triangle Sum Theorem. The angles inside a triangle add up to $180^\circ$. Angles on a straight line also add up to $180^\circ$.
* Step 1: Find the missing interior angle at $F$ (angle $HFG$).
$$180^\circ - (82^\circ + 42^\circ) = 180^\circ - 124^\circ = 56^\circ$$
* Step 2: Find the exterior angle $?$ (angle $GFA$). It forms a linear pair with the $56^\circ$ angle.
$$? = 180^\circ - 56^\circ = 124^\circ$$
*(Alternatively, using the Exterior Angle Theorem: $? = 82^\circ + 42^\circ = 124^\circ$)*
3)
* Concept: Exterior Angle Theorem.
* Calculation: The exterior angle ($123^\circ$) is equal to the sum of the two remote interior angles ($T$ and $S$).
$$123^\circ = ? + 69^\circ$$
Subtract $69$ from both sides:
$$? = 123 - 69$$
$$? = 54^\circ$$
4)
* Concept: Linear Pair and Triangle Sum.
* Step 1: Find the interior angle at $S$ (angle $TSU$). It is supplementary to the $150^\circ$ angle.
$$180^\circ - 150^\circ = 30^\circ$$
* Step 2: Use the Triangle Sum Theorem for triangle $TSU$. We know angle $U$ is $90^\circ$ (right angle symbol) and angle $S$ is $30^\circ$.
$$? + 90^\circ + 30^\circ = 180^\circ$$
$$? + 120^\circ = 180^\circ$$
$$? = 60^\circ$$
5)
* Concept: Linear Pairs and Triangle Sum.
* Step 1: Find interior angle $DCB$. It is supplementary to $49^\circ$.
$$180^\circ - 49^\circ = 131^\circ$$
* Step 2: Find interior angle $DBC$. It is supplementary to $143^\circ$.
$$180^\circ - 143^\circ = 37^\circ$$
* Step 3: Solve for angle $D$ ($?$) using the sum of angles in triangle $DBC$.
$$? + 131^\circ + 37^\circ = 180^\circ$$
$$? + 168^\circ = 180^\circ$$
$$? = 12^\circ$$
6)
* Concept: Exterior Angle Theorem.
* Calculation: The exterior angle ($150^\circ$) equals the sum of the remote interior angles ($34^\circ$ and $?$).
$$150^\circ = 34^\circ + ?$$
Subtract $34$ from both sides:
$$? = 150 - 34$$
$$? = 116^\circ$$
---
7)
* Concept: Exterior Angle Theorem.
* Equation: The exterior angle ($17x - 1$) equals the sum of the remote interior angles ($84$ and $7x + 5$).
$$17x - 1 = 84 + (7x + 5)$$
* Solve:
Combine like terms on the right:
$$17x - 1 = 7x + 89$$
Subtract $7x$ from both sides:
$$10x - 1 = 89$$
Add $1$ to both sides:
$$10x = 90$$
Divide by $10$:
$$x = 9$$
8)
* Concept: Linear Pair and Triangle Sum.
* Step 1: Find the interior angle at $E$ (angle $FEG$). It is supplementary to $130^\circ$.
$$180^\circ - 130^\circ = 50^\circ$$
* Step 2: Set up the equation for the sum of angles in triangle $EFG$.
$$50 + (16x + 4) + (14x + 6) = 180$$
* Solve:
Combine $x$ terms and constant numbers:
$$(16x + 14x) + (50 + 4 + 6) = 180$$
$$30x + 60 = 180$$
Subtract $60$ from both sides:
$$30x = 120$$
Divide by $30$:
$$x = 4$$
9)
* Concept: Exterior Angle Theorem.
* Equation: The exterior angle ($21x + 4$) equals the sum of the remote interior angles ($84$ and $10x - 3$).
$$21x + 4 = 84 + (10x - 3)$$
* Solve:
Simplify the right side:
$$21x + 4 = 10x + 81$$
Subtract $10x$ from both sides:
$$11x + 4 = 81$$
Subtract $4$ from both sides:
$$11x = 77$$
Divide by $11$:
$$x = 7$$
10)
* Concept: Exterior Angle Theorem.
* Equation: The exterior angle ($11x + 3$) equals the sum of the remote interior angles ($85$ and $5x - 10$).
$$11x + 3 = 85 + (5x - 10)$$
* Solve:
Simplify the right side ($85 - 10 = 75$):
$$11x + 3 = 5x + 75$$
Subtract $5x$ from both sides:
$$6x + 3 = 75$$
Subtract $3$ from both sides:
$$6x = 72$$
Divide by $6$:
$$x = 12$$
Final Answer:
1) 94°
2) 124°
3) 54°
4) 60°
5) 12°
6) 116°
7) x = 9
8) x = 4
9) x = 7
10) x = 12
Part 1: Find the measure of each angle indicated
1)
* Concept: The Exterior Angle Theorem states that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
* Calculation: The exterior angle at $C$ (angle $GCD$) is equal to angle $D$ plus angle $E$.
$$? = 70^\circ + 24^\circ$$
$$? = 94^\circ$$
2)
* Concept: Triangle Sum Theorem. The angles inside a triangle add up to $180^\circ$. Angles on a straight line also add up to $180^\circ$.
* Step 1: Find the missing interior angle at $F$ (angle $HFG$).
$$180^\circ - (82^\circ + 42^\circ) = 180^\circ - 124^\circ = 56^\circ$$
* Step 2: Find the exterior angle $?$ (angle $GFA$). It forms a linear pair with the $56^\circ$ angle.
$$? = 180^\circ - 56^\circ = 124^\circ$$
*(Alternatively, using the Exterior Angle Theorem: $? = 82^\circ + 42^\circ = 124^\circ$)*
3)
* Concept: Exterior Angle Theorem.
* Calculation: The exterior angle ($123^\circ$) is equal to the sum of the two remote interior angles ($T$ and $S$).
$$123^\circ = ? + 69^\circ$$
Subtract $69$ from both sides:
$$? = 123 - 69$$
$$? = 54^\circ$$
4)
* Concept: Linear Pair and Triangle Sum.
* Step 1: Find the interior angle at $S$ (angle $TSU$). It is supplementary to the $150^\circ$ angle.
$$180^\circ - 150^\circ = 30^\circ$$
* Step 2: Use the Triangle Sum Theorem for triangle $TSU$. We know angle $U$ is $90^\circ$ (right angle symbol) and angle $S$ is $30^\circ$.
$$? + 90^\circ + 30^\circ = 180^\circ$$
$$? + 120^\circ = 180^\circ$$
$$? = 60^\circ$$
5)
* Concept: Linear Pairs and Triangle Sum.
* Step 1: Find interior angle $DCB$. It is supplementary to $49^\circ$.
$$180^\circ - 49^\circ = 131^\circ$$
* Step 2: Find interior angle $DBC$. It is supplementary to $143^\circ$.
$$180^\circ - 143^\circ = 37^\circ$$
* Step 3: Solve for angle $D$ ($?$) using the sum of angles in triangle $DBC$.
$$? + 131^\circ + 37^\circ = 180^\circ$$
$$? + 168^\circ = 180^\circ$$
$$? = 12^\circ$$
6)
* Concept: Exterior Angle Theorem.
* Calculation: The exterior angle ($150^\circ$) equals the sum of the remote interior angles ($34^\circ$ and $?$).
$$150^\circ = 34^\circ + ?$$
Subtract $34$ from both sides:
$$? = 150 - 34$$
$$? = 116^\circ$$
---
Part 2: Solve for x
7)
* Concept: Exterior Angle Theorem.
* Equation: The exterior angle ($17x - 1$) equals the sum of the remote interior angles ($84$ and $7x + 5$).
$$17x - 1 = 84 + (7x + 5)$$
* Solve:
Combine like terms on the right:
$$17x - 1 = 7x + 89$$
Subtract $7x$ from both sides:
$$10x - 1 = 89$$
Add $1$ to both sides:
$$10x = 90$$
Divide by $10$:
$$x = 9$$
8)
* Concept: Linear Pair and Triangle Sum.
* Step 1: Find the interior angle at $E$ (angle $FEG$). It is supplementary to $130^\circ$.
$$180^\circ - 130^\circ = 50^\circ$$
* Step 2: Set up the equation for the sum of angles in triangle $EFG$.
$$50 + (16x + 4) + (14x + 6) = 180$$
* Solve:
Combine $x$ terms and constant numbers:
$$(16x + 14x) + (50 + 4 + 6) = 180$$
$$30x + 60 = 180$$
Subtract $60$ from both sides:
$$30x = 120$$
Divide by $30$:
$$x = 4$$
9)
* Concept: Exterior Angle Theorem.
* Equation: The exterior angle ($21x + 4$) equals the sum of the remote interior angles ($84$ and $10x - 3$).
$$21x + 4 = 84 + (10x - 3)$$
* Solve:
Simplify the right side:
$$21x + 4 = 10x + 81$$
Subtract $10x$ from both sides:
$$11x + 4 = 81$$
Subtract $4$ from both sides:
$$11x = 77$$
Divide by $11$:
$$x = 7$$
10)
* Concept: Exterior Angle Theorem.
* Equation: The exterior angle ($11x + 3$) equals the sum of the remote interior angles ($85$ and $5x - 10$).
$$11x + 3 = 85 + (5x - 10)$$
* Solve:
Simplify the right side ($85 - 10 = 75$):
$$11x + 3 = 5x + 75$$
Subtract $5x$ from both sides:
$$6x + 3 = 75$$
Subtract $3$ from both sides:
$$6x = 72$$
Divide by $6$:
$$x = 12$$
Final Answer:
1) 94°
2) 124°
3) 54°
4) 60°
5) 12°
6) 116°
7) x = 9
8) x = 4
9) x = 7
10) x = 12
Parent Tip: Review the logic above to help your child master the concept of triangle sum and exterior angle theorem worksheet.
![ANSWERED] Worksheet Triangle Sum and Exterior angle Theorem I Find ...](/images2/triangle-sum-and-exterior-angle-theorem-worksheet/triangle-sum-and-exterior-angle-theorem-worksheet-28.jpg)
