Answer: " x = 35 " .
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Step-by-step explanation:
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We are asked to find the value of "x" .
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From the diagram, we see that
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→ ∡DEA and ∡AEC — together — form a straight line.
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Note that by definition:
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1) All angles that form a straight line (or "line segment") — are "supplementary angles" ; And :
2) All supplementary angles add up to 180° .
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As such:
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Since:
∡DEA and ∡AEC— together — form a straight line ;
→ ∡DEA and ∡AEC are supplementary angles ;
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So: m∠DEA + m∠AEC = 180 .
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So, we plug in our known (given) values for:
→ " m∠DEA " ; and: " m∠AEC " ; and rewrite the equation:
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→ 4x + 40 = 180 ; Find the value of "x" .
Subtract "40 from each side of the equation:
→ 4x + 40 − 40 = 180 − 40 ;
to get:
→ 4x = 140 .
Now, divide each side of the equation by "4" ;
to isolate "x" on one side of the equation;
& to solve for "x" :
→ 4x / 4 = 140 / 4 ;
to get:
→ " x = 35 " ; → which is our answer.
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Note: Let's confirm our answer—by using another method to solve the problem.
Note: ∡AEC and ∡CEB —together—form a straight line and thus, by definition, are supplementary angles that add up to 180° [as aforementioned.].
We are given (from image): m∠AEC = 40 ;
& we need to solve for (m∠CEB).
→ (m∠CEB) + (m∠AEC) = 180 ;
Now, plug in "40" for "(m∠AEC)" ; & rewrite the equation:
→ (m∠CEB) + 40 = 180 ;
Subtract "40" from each side of the equation ;
to isolate: (m∠CEB) on one side of the equation;
& to solve for (m∠CEB) ;
→ (m∠CEB) + 40 − 40 = 180 − 40 ;
to get: m∠CEB = 140.
Does this makes sense?
→ m∠AEC + 140 =? 180 ?? ; → 40 + 140 =? 180 ?? ; 180 =? 180?? Yes!
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Now, notice from the "image", that ∡CEB and ∡AED are "vertical angles".
Note that "vertical angles" are congruent (have the same measurements); as such:
→ m∠CEB = m∠AED ; Note: m∠CEB = 140 [just calculated] ;
m∠AED = 4x [given w/in "image"] ;
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So;
→ m∠CEB = m∠AED ;
→ Rewrite as: 140 = 4x ;
↔ 4x = 140 ; Solve for "x" .
Now, divide each side of the equation by "4" ;
to isolate "x" on one side of the equation;
& to solve for "x" :
→ 4x / 4 = 140 / 4 ;
to get:
→ " x = 35 " ; → which is our answer ; which is the same value for "x" as previously calculated!
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Hope this answer—and rather lengthy explanation is helpful!
Best wishes!
