Steps and solution---Question 10:
Lines d₁ and d₂ are parallel. What is the value of the 3x?
Options:
A) 60
B) 70
C) 80
D) 90
Chart Description:
Type: Geometric diagram showing two parallel lines intersected by a transversal.
Main Elements:
- Two horizontal lines labeled d₁ and d₂. Arrows at the ends of the lines indicate they are lines, potentially extending infinitely. The question states d₁ and d₂ are parallel.
- A single transversal line intersects both parallel lines.
- Angles are labeled around the intersection points:
- An angle between line d₁ and the transversal is labeled 40°. This angle is on the upper side of d₁ and to the left of the transversal intersection point.
- An angle is formed by the transversal between the parallel lines, labeled 3x°. This angle forms a 'zig-zag' shape with the 40° and 50° angles.
- An angle between line d₂ and the transversal is labeled 50°. This angle is on the lower side of d₂ and to the left of the transversal intersection point. The 50° angle is below the 3x° angle.
视频信息
答案文本
视频字幕
We have two parallel lines d₁ and d₂ intersected by a transversal. We need to find the value of 3x. We can see a 40-degree angle at the upper intersection and a 50-degree angle at the lower intersection, with angle 3x between the parallel lines.
To solve this problem, we draw an auxiliary line through the vertex of angle 3x, parallel to both d₁ and d₂. This auxiliary line divides the angle 3x into two parts. The upper part equals 40 degrees and the lower part equals 50 degrees due to alternate interior angle relationships.
Using the property of alternate interior angles, we can determine that the upper part of angle 3x equals 40 degrees, and the lower part equals 50 degrees. These angles are equal to their corresponding alternate interior angles formed by the transversal and the parallel lines.
Now we can calculate the total angle. Since angle 3x is the sum of the upper part (40 degrees) and the lower part (50 degrees), we have 3x equals 40 plus 50, which equals 90 degrees. Therefore, the answer is D) 90.
To summarize our solution: We drew an auxiliary line parallel to the given parallel lines, which divided angle 3x into two parts. Using the alternate interior angles property, we found that the upper part equals 40 degrees and the lower part equals 50 degrees. Therefore, 3x equals 90 degrees, making the answer D) 90. This problem demonstrates the important geometric principle that alternate interior angles are equal when parallel lines are cut by a transversal.