Solved on Dec 15, 2023

Given congruent triangles $\triangle ABC \cong \triangle DEF$ with $AB=4x-5$ , $DE=67$ , $\angle C=5y+7$ , and $\angle F=57$ , find $x$ and $y$ .

STEP 1

Assumptions
1. $\triangle ABC \cong \triangle DEF$ implies that corresponding sides and angles are equal.
2. $AB = 4x - 5$ corresponds to $DE = 67$ .
3. $m\angle C = 5y + 7$ corresponds to $m\angle F = 57$ .
4. We need to find the values of $x$ and $y$ .

STEP 2

Since the triangles are congruent, corresponding sides are equal. Therefore, we can set the expression for $AB$ equal to $DE$ .
$AB = DE$

STEP 3

Substitute the given expressions for $AB$ and $DE$ .
$4x - 5 = 67$

STEP 4

Solve for $x$ by adding 5 to both sides of the equation.
$4x - 5 + 5 = 67 + 5$

STEP 5

Simplify the equation.
$4x = 72$

STEP 6

Divide both sides by 4 to isolate $x$ .
$\frac{4x}{4} = \frac{72}{4}$

STEP 7

Calculate the value of $x$ .
$x = 18$

STEP 8

Since the triangles are congruent, corresponding angles are equal. Therefore, we can set the expression for $m\angle C$ equal to $m\angle F$ .
$m\angle C = m\angle F$

STEP 9

Substitute the given expressions for $m\angle C$ and $m\angle F$ .
$5y + 7 = 57$

STEP 10

Solve for $y$ by subtracting 7 from both sides of the equation.
$5y + 7 - 7 = 57 - 7$

STEP 11

Simplify the equation.
$5y = 50$

STEP 12

Divide both sides by 5 to isolate $y$ .
$\frac{5y}{5} = \frac{50}{5}$

STEP 13

Calculate the value of $y$ .
$y = 10$
The values of $x$ and $y$ are $x = 18$ and $y = 10$ .

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Given congruent triangles $\triangle ABC \cong \triangle DEF$ with $AB=4x-5$ , $DE=67$ , $\angle C=5y+7$ , and $\angle F=57$ , find $x$ and $y$ .

STEP 1

Assumptions
1. $\triangle ABC \cong \triangle DEF$ implies that corresponding sides and angles are equal.
2. $AB = 4x - 5$ corresponds to $DE = 67$ .
3. $m\angle C = 5y + 7$ corresponds to $m\angle F = 57$ .
4. We need to find the values of $x$ and $y$ .

STEP 2

Since the triangles are congruent, corresponding sides are equal. Therefore, we can set the expression for $AB$ equal to $DE$ .
$AB = DE$

STEP 3

Substitute the given expressions for $AB$ and $DE$ .
$4x - 5 = 67$

STEP 4

Solve for $x$ by adding 5 to both sides of the equation.
$4x - 5 + 5 = 67 + 5$

STEP 5

Simplify the equation.
$4x = 72$

STEP 6

Divide both sides by 4 to isolate $x$ .
$\frac{4x}{4} = \frac{72}{4}$

STEP 7

Calculate the value of $x$ .
$x = 18$

STEP 8

Since the triangles are congruent, corresponding angles are equal. Therefore, we can set the expression for $m\angle C$ equal to $m\angle F$ .
$m\angle C = m\angle F$

STEP 9

Substitute the given expressions for $m\angle C$ and $m\angle F$ .
$5y + 7 = 57$

STEP 10

Solve for $y$ by subtracting 7 from both sides of the equation.
$5y + 7 - 7 = 57 - 7$

STEP 11

Simplify the equation.
$5y = 50$

STEP 12

Divide both sides by 5 to isolate $y$ .
$\frac{5y}{5} = \frac{50}{5}$

STEP 13

Calculate the value of $y$ .
$y = 10$
The values of $x$ and $y$ are $x = 18$ and $y = 10$ .

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Given congruent triangles △ABC≅△DEF\triangle ABC \cong \triangle DEF△ABC≅△DEF with AB=4x−5AB=4x-5AB=4x−5, DE=67DE=67DE=67, ∠C=5y+7\angle C=5y+7∠C=5y+7, and ∠F=57\angle F=57∠F=57, find xxx and yyy.

STEP 1

STEP 2

Since the triangles are congruent, corresponding sides are equal. Therefore, we can set the expression for ABABAB equal to DEDEDE.AB=DEAB = DEAB=DE

STEP 3

Substitute the given expressions for ABABAB and DEDEDE.4x−5=674x - 5 = 674x−5=67

STEP 4

Solve for xxx by adding 5 to both sides of the equation.4x−5+5=67+54x - 5 + 5 = 67 + 54x−5+5=67+5

STEP 5

Simplify the equation.4x=724x = 724x=72

STEP 6

Divide both sides by 4 to isolate xxx.4x4=724\frac{4x}{4} = \frac{72}{4}44x​=472​

STEP 7

Calculate the value of xxx.x=18x = 18x=18

STEP 8

Since the triangles are congruent, corresponding angles are equal. Therefore, we can set the expression for m∠Cm\angle Cm∠C equal to m∠Fm\angle Fm∠F.m∠C=m∠Fm\angle C = m\angle Fm∠C=m∠F

STEP 9

Substitute the given expressions for m∠Cm\angle Cm∠C and m∠Fm\angle Fm∠F.5y+7=575y + 7 = 575y+7=57

STEP 10

Solve for yyy by subtracting 7 from both sides of the equation.5y+7−7=57−75y + 7 - 7 = 57 - 75y+7−7=57−7

STEP 11

Simplify the equation.5y=505y = 505y=50

STEP 12

Divide both sides by 5 to isolate yyy.5y5=505\frac{5y}{5} = \frac{50}{5}55y​=550​

STEP 13

Calculate the value of yyy.y=10y = 10y=10The values of xxx and yyy are x=18x = 18x=18 and y=10y = 10y=10.

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Given congruent triangles $\triangle ABC \cong \triangle DEF$ with $AB=4x-5$ , $DE=67$ , $\angle C=5y+7$ , and $\angle F=57$ , find $x$ and $y$ .

Since the triangles are congruent, corresponding sides are equal. Therefore, we can set the expression for $AB$ equal to $DE$ .
$AB = DE$

Substitute the given expressions for $AB$ and $DE$ .
$4x - 5 = 67$

Solve for $x$ by adding 5 to both sides of the equation.
$4x - 5 + 5 = 67 + 5$

Simplify the equation.
$4x = 72$

Divide both sides by 4 to isolate $x$ .
$\frac{4x}{4} = \frac{72}{4}$

Calculate the value of $x$ .
$x = 18$

Since the triangles are congruent, corresponding angles are equal. Therefore, we can set the expression for $m\angle C$ equal to $m\angle F$ .
$m\angle C = m\angle F$

Substitute the given expressions for $m\angle C$ and $m\angle F$ .
$5y + 7 = 57$

Solve for $y$ by subtracting 7 from both sides of the equation.
$5y + 7 - 7 = 57 - 7$

Simplify the equation.
$5y = 50$

Divide both sides by 5 to isolate $y$ .
$\frac{5y}{5} = \frac{50}{5}$

Calculate the value of $y$ .
$y = 10$
The values of $x$ and $y$ are $x = 18$ and $y = 10$ .