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Angles
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Angles: Very Difficult Problems with Solutions
By Catalin David
Problem 1
The sides of two angles are parallel two by two. If one of them has a measure of 45°, what is the measure of the other one?
60°
45°
120°
65°
Solution:
We make line EF longer until it intersecta line AB in point M. ∠DEF and ∠AME are corresponding. Since DE and AM are parallel, m∠AMF also has 45°. ∠AME and ∠ABC are corresponding. Since MF and BC are parallel, ∠ABC also has 45°. To conclude, two acute angles whose sides are parallel two by two have the same measure.
Problem 2
The sides of two obtuse angles are parallel two by two. If one of them has 130°, what is the measure of the other angle?
150°
60°
130°
120°
Solution:
We make side EF longer until it intersects side AB in point M. ∠DEF and ∠AME are corresponding. Since DE and AM are parallel, ∠AMF also has 130°. ∠AME and ∠ABC are corresponding. Since MF and BC are parallel, ∠ABC also has 130°. To conclude, two angles whose sides are parallel two by two have the same measure.
Problem 3
Two angles, one acute and one obtuse have sides which are parallel two by two. If one of them has 40°, what is the measure of the other one?
120°
40°
140°
160°
Solution:
We make EF longer until it intersects AB at point M. ∠DEF and ∠AMN are corresponding. Since DE and AM are parallel, ∠AMN and ∠DEF are equal. ∠AME and ∠ABC are corresponding. Since MF and BC are parallel, m∠AME = 40°. ∠AMN and ∠AME are adjacent and supplementary. Thus, m∠AMN=140°, so m∠DEF=140°. Two conclude, two angles, one of them acute and the other obtuse, whose sides are parallel, are supplementary.
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