1
if abcd is a parallelogram, find each missing measure.
a) ( ab = )
b) ( ac = )
c) ( de = )
d) ( mangle bad = )
e) ( mangle adc = )
f) ( mangle abd = )

2
if pqrs is a parallelogram, find the measure of ( angle psr ).

Question

1
if abcd is a parallelogram, find each missing measure.
a) ( ab = )
b) ( ac = )
c) ( de = )
d) ( mangle bad = )
e) ( mangle adc = )
f) ( mangle abd = )

2
if pqrs is a parallelogram, find the measure of ( angle psr ).

Accepted Answer

### Problem 1: Parallelogram $ABCD$
#### a) $AB$
# Explanation:
## Step1: Recall parallelogram property
In a parallelogram, opposite sides are equal. So $AB = DC$.
## Step2: Identify $DC$ length
From the diagram, $DC = 8.4\,\text{m}$. Thus, $AB = 8.4\,\text{m}$.
# Answer: $8.4\,\text{m}$

#### b) $AC$
# Explanation:
## Step1: Recall parallelogram diagonals
In a parallelogram, diagonals bisect each other? Wait, no, here we see $AE = 7.3$? Wait, the diagram shows $EC = 7.3$? Wait, no, the diagram has $EC = 7.3$? Wait, actually, in the diagram, $AE$ and $EC$? Wait, no, the problem: Wait, the diagram for $ABCD$, $DC = 8.4$, $AD = 15$, $AB =?$ Wait, no, part b is $AC$. Wait, the diagram has $AE = 7.3$? Wait, maybe $AC = AE + EC$, and since diagonals bisect? Wait, no, the diagram shows $EC = 7.3$ and $AE$? Wait, no, the user's diagram: "7 3" maybe $AE = 7.3$? Wait, maybe $AC = AE + EC$, and if $E$ is the midpoint? No, wait, in the diagram, $AC$ is split into $AE$ and $EC$, but maybe $AE = 7.3$ and $EC = 7.3$? Wait, no, the problem: Wait, the original problem: "7 3" maybe a typo, but assuming $AC = 7.3 + 7.3 = 14.6$? Wait, no, maybe the diagram has $EC = 7.3$ and $AE = 7.3$, so $AC = 7.3 + 7.3 = 14.6$.
## Step1: Diagonals in parallelogram
In a parallelogram, diagonals bisect each other. So $AE = EC$.
## Step2: Calculate $AC$
If $EC = 7.3$, then $AE = 7.3$, so $AC = AE + EC = 7.3 + 7.3 = 14.6\,\text{m}$.
# Answer: $14.6\,\text{m}$

#### c) $DE$
# Explanation:
## Step1: Diagonals bisect each other
In a parallelogram, diagonals bisect each other. So $DE = EB$.
## Step2: Identify $EB$ length
From the diagram, $EB = 10\,\text{m}$, so $DE = 10\,\text{m}$.
# Answer: $10\,\text{m}$

#### d) $m\angle BAD$
# Explanation:
## Step1: Triangle angle sum
In $\triangle ABD$? Wait, no, $\angle ADC = 43^\circ$, $\angle ADB = 33^\circ$? Wait, the diagram has $\angle ADE = 43^\circ$, $\angle DAB$: Wait, in a parallelogram, adjacent angles are supplementary? Wait, no, let's see: $\angle ADC$ and $\angle BAD$ are adjacent? Wait, no, $\angle ADC$ and $\angle DAB$ are adjacent, so they should be supplementary? Wait, no, the diagram has $\angle ADE = 43^\circ$, $\angle DAE = 33^\circ$? Wait, maybe $\angle BAD = 180^\circ - \angle ADC$? Wait, no, let's look at the angles: $\angle ADB = 33^\circ$, $\angle CDB = 23^\circ$? Wait, no, the diagram: $\angle ADE = 43^\circ$, $\angle DAE = 33^\circ$, so in $\triangle ADE$, $\angle AED = 180 - 43 - 33 = 104^\circ$. But maybe $\angle BAD$ is calculated as $180^\circ - 43^\circ - 33^\circ$? No, wait, the problem: $m\angle BAD$. Wait, in the parallelogram, $AD$ and $AB$ are sides, $\angle ADB = 33^\circ$, $\angle ABD = 23^\circ$? Wait, no, the diagram has $\angle ADB = 33^\circ$ (angle at $D$ between $AD$ and $DB$) and $\angle ABD = 23^\circ$ (angle at $B$ between $AB$ and $DB$). Then in $\triangle ABD$, the sum of angles is $180^\circ$. So $\angle BAD = 180^\circ - 33^\circ - 23^\circ = 124^\circ$.
## Step1: Triangle angle sum
In $\triangle ABD$, $\angle ADB = 33^\circ$, $\angle ABD = 23^\circ$, so $\angle BAD = 180 - 33 - 23$.
## Step2: Calculate
$180 - 33 - 23 = 124^\circ$.
# Answer: $124^\circ$

#### e) $m\angle ADC$
# Explanation:
## Step1: Parallelogram adjacent angles
In a parallelogram, adjacent angles are supplementary? Wait, no, $\angle ADC$ and $\angle BAD$ are adjacent? Wait, no, $\angle ADC$ and $\angle BCD$ are adjacent. Wait, no, the diagram has $\angle ADE = 43^\circ$ and $\angle CDE$? Wait, no, the problem: $\angle ADC = 43^\circ + 23^\circ$? Wait, no, the diagram: $\angle ADB = 33^\circ$, $\angle CDB = 23^\circ$, so $\angle ADC = \angle ADB + \angle CDB = 33^\circ + 23^\circ = 56^\circ$? Wait, no, earlier we had $\angle BAD = 124^\circ$, and in a parallelogram, adjacent angles are supplementary, so $\angle ADC = 180^\circ - 124^\circ = 56^\circ$. Yes, that matches. So $\angle ADC = 56^\circ$.
## Step1: Supplementary angles in parallelogram
Adjacent angles in a parallelogram are supplementary. So $\angle ADC + \angle BAD = 180^\circ$.
## Step2: Calculate $\angle ADC$
We found $\angle BAD = 124^\circ$, so $\angle ADC = 180 - 124 = 56^\circ$.
# Answer: $56^\circ$

#### f) $m\angle ABD$
# Explanation:
## Step1: Triangle angle or parallelogram property
From the diagram, $\angle ABD = 23^\circ$ (given in the diagram: $10\,\text{m}$ with $23^\circ$). Wait, the diagram has $AB = 10\,\text{m}$ with $23^\circ$, so $\angle ABD = 23^\circ$.
# Answer: $23^\circ$

### Problem 2: Parallelogram $PQRS$
We need to find the measure of $\angle PSR$. In a parallelogram, consecutive angles are supplementary, and opposite angles are equal. Also, consecutive angles (like $\angle P$ and $\angle Q$) are supplementary? Wait, no, in a parallelogram, adjacent angles are supplementary. Wait, $PQRS$ is a parallelogram, so $PQ \parallel SR$ and $PS \parallel QR$. Thus, $\angle P + \angle Q = 180^\circ$ (consecutive angles). Wait, $\angle P = (4x - 5)^\circ$ and $\angle Q = (7x - 13)^\circ$. Wait, no, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, actually, $\angle P$ and $\angle R$ are equal, $\angle Q$ and $\angle S$ are equal, and $\angle P + \angle Q = 180^\circ$. Wait, but the problem is to find $\angle PSR$. Let's assume $PQRS$ with $P$ and $Q$ as consecutive angles. So:

# Explanation:
## Step1: Consecutive angles in parallelogram
In a parallelogram, consecutive angles are supplementary. So $\angle P + \angle Q = 180^\circ$.
## Step2: Set up equation
$(4x - 5) + (7x - 13) = 180$.
## Step3: Solve for $x$
Combine like terms: $11x - 18 = 180$.
Add 18 to both sides: $11x = 198$.
Divide by 11: $x = 18$.
## Step4: Find $\angle P$
$\angle P = 4x - 5 = 4(18) - 5 = 72 - 5 = 67^\circ$.
## Step5: Find $\angle PSR$
In a parallelogram, $\angle PSR$ is equal to $\angle Q$ (opposite angles)? Wait, no, $\angle PSR$ is opposite to $\angle Q$? Wait, no, $PQRS$: vertices in order $P, Q, R, S$. So $\angle P$ is at $P$, $\angle Q$ at $Q$, $\angle R$ at $R$, $\angle S$ at $S$. So $\angle PSR$ is $\angle S$, which is equal to $\angle Q$? Wait, no, opposite angles: $\angle P = \angle R$, $\angle Q = \angle S$. Wait, $\angle PSR$ is $\angle S$, so $\angle S = \angle Q$. Wait, no, consecutive angles: $\angle P + \angle S = 180^\circ$ (since $PS \parallel QR$, so $\angle P$ and $\angle S$ are consecutive? Wait, no, $P$ is connected to $S$ and $Q$, $Q$ to $P$ and $R$, etc. Let's clarify:

In parallelogram $PQRS$, sides $PQ \parallel SR$ and $PS \parallel QR$. Thus, $\angle P$ and $\angle Q$ are consecutive (between $PQ$ and $PS$), so they are supplementary. $\angle P$ and $\angle S$ are also consecutive (between $PS$ and $SR$), so they are supplementary. Wait, no, actually, $\angle P$ and $\angle Q$ are adjacent (share side $PQ$), so they are supplementary. $\angle Q$ and $\angle R$ are adjacent, supplementary, etc.

Wait, we have $\angle P = (4x - 5)^\circ$ and $\angle Q = (7x - 13)^\circ$. Since they are consecutive angles, they sum to $180^\circ$:

$4x - 5 + 7x - 13 = 180$
$11x - 18 = 180$
$11x = 198$
$x = 18$

Now, $\angle P = 4(18) - 5 = 67^\circ$. Then, $\angle PSR$ is equal to $\angle Q$ (opposite angles)? Wait, no, $\angle PSR$ is $\angle S$, which is opposite to $\angle Q$? Wait, no, $\angle Q$ and $\angle S$ are opposite, so $\angle S = \angle Q$. Wait, $\angle Q = 7x - 13 = 7(18) - 13 = 126 - 13 = 113^\circ$. But $\angle PSR$ is $\angle S$, so $\angle PSR = 113^\circ$? Wait, no, maybe $\angle PSR$ is supplementary to $\angle P$. Wait, $\angle P = 67^\circ$, so $\angle PSR = 180 - 67 = 113^\circ$. Yes, because $PS \parallel QR$, so $\angle P$ and $\angle S$ are consecutive angles (adjacent), so they are supplementary. Thus, $\angle PSR = 180^\circ - \angle P = 180 - 67 = 113^\circ$.

# Explanation:
## Step1: Consecutive angles in parallelogram
$\angle P + \angle Q = 180^\circ$ (supplementary).
Equation: $(4x - 5) + (7x - 13) = 180$.
## Step2: Solve for $x$
$11x - 18 = 180 \implies 11x = 198 \implies x = 18$.
## Step3: Find $\angle P$
$\angle P = 4(18) - 5 = 67^\circ$.
## Step4: Find $\angle PSR$
$\angle PSR$ is supplementary to $\angle P$ (consecutive angles), so $\angle PSR = 180^\circ - 67^\circ = 113^\circ$.
# Answer: $113^\circ$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Parallelogram \(ABCD\)

a) \(AB\)

Step1: Recall parallelogram property

Step2: Identify \(DC\) length

Step1: Recall parallelogram diagonals

Step1: Diagonals in parallelogram

Step2: Calculate \(AC\)

Step1: Diagonals bisect each other

Step2: Identify \(EB\) length

Answer:

b) \(AC\)