Question

complete the proof that m∠t + m∠x + m∠tux = 180° without using the triangle angle sum theorem.

1. (overline{tx}paralleloverline{vw})
2. (angle tcongangle vuw)
3. (angle xcongangle wux)
4. m∠vux = m∠vuw + m∠wux
5. m∠vux = m∠t + m∠x
6. m∠vux + m∠tux = 180°
7. m∠t + m∠x + m∠tux = 180°

Explanation:

Step1: Given information

$\overline{TX}\parallel\overline{VW}$

Step2: Alternate - interior angles

$\angle X\cong\angle WUX$ (Alternate - interior angles are congruent when two parallel lines are cut by a transversal)

Step3: Given information

$\overline{TU}\cong\overline{VU}$

Step4: Base - angle theorem

$\angle T\cong\angle V$ (Base - angle theorem: In an isosceles triangle, the base angles are congruent)

Step5: Angle - addition postulate

$m\angle VUX=m\angle VUW + m\angle WUX$

Step6: Substitution

Since $m\angle X = m\angle WUX$ and $m\angle VUX=m\angle VUW + m\angle WUX$, and $m\angle T=m\angle V$, we have $m\angle VUX + m\angle T+ m\angle X=m\angle T + m\angle VUW + m\angle WUX+m\angle X$

Step7: Linear - pair property

$\angle VUX$ and $\angle TUX$ form a linear pair, so $m\angle VUX + m\angle TUX = 180^{\circ}$. Also, $m\angle T + m\angle X+m\angle TUX=m\angle T + m\angle VUW + m\angle WUX+m\angle X$ (by substitution). So $m\angle T + m\angle X+m\angle TUX = 180^{\circ}$

Answer:

The proof is completed as above. The reasons for each statement are: Alternate - interior angles for step 2, Base - angle theorem for step 4, Angle - addition postulate for step 5, Substitution for step 6 and 7, and Linear - pair property for step 7.