Answer:
1)∠DOF, with F (-5,-1) 2) x=30º y=90º 3)∠x=∠y=50º 4) a)x=35º b=110º 5) x=14 y=15 6) ∠EIG≅FIH and ∠EIF ≅ ∠GIH 7) True because of ∠ABC=∠XYZ
Step-by-step explanation:
1) Supplementary angles: two angles whose sum is equivalent to 180º. Adjacent angles share the same vertex. So DOF measures 135º.
2) Vertical Angles Theorem assures us that 2x= y-x. Moreover to that, 2x+(y+x) =180º And (y+x)+(y-x)=180º
I) 2x=y-x ∴3x=y
II) (y+x)+(y-x)=180º If 2y=180º → y =90º
If 3x=y and 3x=90 → x=30
3) By the Vertical Angle theorem, two angles sharing the same vertex and opposite are congruent. Also, the ∠x +40º are complementary angles, i.e. their sum is equal to 90º.
Then x =y
So ∠x=∠y=50º
4)
a) Similarly to the 1st case:
y=2x (Vertical Angles Theorem)
2x+(x+y+5)=180 (Supplementary Angles)
Algebraically.
Then, let's solve a Linear System of Equations.
2x-y=0
3x+y=175
----------------
5x =175 ∴ x=35
If y=2x Then y=70º
b) The Unlabeled angle it's equal to (x+y+5) according to Vertical Angle Theorem
So 35+70+5=105+5=110º
5) Similarly to 1st case, and to the previous one:
a)1) 3x+8=5x-20
2) 3x+8+5x+4y=180 ∴ 8x +4y=172
-2x+28=0 *4
8x+4y=172
--------------------
-8x+0=-112
8x+4y=172
-----------------
4y=60
y=15 and 8x +60=172 ∴ 8x=112 x=14
6)
∠EIG≅FIH and ∠EIF ≅ ∠GIH For the first both pairs share the same vertex, the first one is perpendicular and the second one by the Vertical Theorem.
7) If Supplementary angles the first one measures the same as the second one then ∠ABC=∠XYZ = So we can say that's true, since the only positive integers, with the same value that sums up to 180º, is 90º + 90º
