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Difficulty: MediumLines and Angles

In a geometric plane, line L1L_1 is parallel to line L2L_2. Points AA and CC lie on line L1L_1, and points BB and DD lie on line L2L_2. Line segments ABAB and CDCD intersect at point XX located between lines L1L_1 and L2L_2. If measure of ∠XAC=42∘\angle XAC = 42^\circ and measure of ∠XDB=35∘\angle XDB = 35^\circ, what is the measure, in degrees, of ∠AXC\angle AXC?

Answer: 103 degrees

Answer

The measure of ∠AXC\angle AXC is 103∘103^\circ.
Line L1L_1 is parallel to line L2L_2, meaning segment ACAC is parallel to segment BDBD. Transversal line CDCD intersects both parallel lines, creating alternate interior angles ∠XCA\angle XCA and ∠XDB\angle XDB. Hence, ∠XCA=∠XDB=35∘\angle XCA = \angle XDB = 35^\circ. Inside triangle ACXACX, the three interior angles must sum to 180∘180^\circ. Substituting the values yields ∠AXC=180∘−(42∘+35∘)=103∘\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ.

Step-by-Step Solution

1
Identify parallel lines and the transversal line
Line segment CDCD acts as a transversal line intersecting parallel lines L1L_1 and L2L_2.
Points AA and CC lie on line L1L_1 while points BB and DD lie on line L2L_2 with L1∥L2L_1 \parallel L_2.
2
Apply the alternate interior angles theorem
\angle XCA = \angle XDB = 35^\circ
When a transversal intersects two parallel lines, alternate interior angles are equal.
3
Calculate the target angle using the sum of interior angles in a triangle
\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ
The sum of interior angles in triangle ACXACX is 180∘180^\circ.

Key Concept

Properties of parallel lines intersected by a transversal and the triangle angle sum theorem.
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Topics

Lines and Angles

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