a m∠ BAE=y^(∘) m∠ DAE=180^(∘)-y^(∘) m∠ CAB=180^(∘)-y^(∘)
B
b See solution.
Practice makes perfect
a Let's assume that CE and BD are straight lines. The ray AD is in the interior of ∠ CAE, which according to our assumption is a straight angle. Therefore, ∠ CAD and ∠ DAE form a linear pair and are supplementary.
m∠ CAD + m∠ DAE = 180^(∘)
Using that m∠ CAD=y^(∘), we can rewrite this equation.
y^(∘) + m∠ DAE = 180^(∘) ⇒ m∠ DAE =180^(∘) - y^(∘)
The ray AC is in the interior of ∠ BAD. We assumed that BD is a straight line. Therefore ∠ CAB and ∠ CAD must also be a linear pair and supplementary.
m∠ CAB + m∠ CAD =180^(∘)
Again we use that m∠ CAD=y^(∘). We can now rewrite our equation.
m∠ CAB + y^(∘) =180^(∘) ⇒ m∠ CAB =180^(∘) - y^(∘)
The ray AB is in the interior of ∠ CAE, which according to our assumption is a straight angle.
m∠ CAB + ∠ BAE =180^(∘)
Let's now rewrite this using that we know m∠ CAB =180^(∘) - y^(∘).
180^(∘) - y^(∘) + ∠ BAE =180^(∘) ⇒ ∠ BAE =y^(∘)
To summarise we can write the three angles as follows.
&∠ BAE=y^(∘)
&∠ DAE=180^(∘)-y^(∘)
&∠ CAB=180^(∘)-y^(∘)
b ∠ BAE and ∠ CAD are vertical angles and the two angles have the same measure — they are congruent. ∠ CAB and the ∠ DAE are also a pair of vertical angles. These two angles also have the same measure, making them congruent. We notice that if two angles are vertical angles they are also congruent.
