Example Question - corresponding angles
Here are examples of questions we've helped users solve.
Identifying the Correct Statement to Prove Parallel Lines in a Geometry Problem
\begin{align*} & \text{To prove that } \overline{ABCD} || \overline{EFG}, \\ & \text{the corresponding angles must be equal, i.e.,} \\ & \angle BFC = \angle GFC. \\ & \text{Thus, the correct statement is option (4).} \end{align*}
Proving Similarity of Triangles with Parallel Lines
Để chứng minh rằng hai tam giác \(\triangle ABD\) và \(\triangle CEB\) đồng dạng với nhau, chúng ta cần chứng minh được rằng chúng có tỉ lệ các cạnh tương ứng bằng nhau và/hoặc các góc tương ứng bằng nhau. Trong hình vẽ có tam giác \(\triangle ABD\) và \(\triangle CEB\) với \(AB\) là cạnh chung và hai đường chéo \(BD\) và \(CE\) cắt nhau tại \(A\). Chúng ta có thể sử dụng tiên đề đường thẳng song song để chứng minh hai tam giác này đồng dạng. Cụ thể, từ hình vẽ, ta nhận thấy rằng \(AD\) song song với \(BE\). Khi hai đường thẳng song song được cắt bởi một đường chéo (ở đây là \(AB\)), thì các góc đồng vị và các góc so le trong sẽ bằng nhau: \(\angle ADB = \angle BEC\) và \(\angle DAB = \angle EBC\). Vì vậy, ta có các cặp góc tương ứng bằng nhau: - \(\angle ABD\) (chung cho cả hai tam giác) - \(\angle ADB = \angle BEC\) (vì \(AD \parallel BE\)) - \(\angle BDA\) khác với \(\angle BEC\) nhưng \(\angle BAD\) tương ứng với \(\angle BCE\) Với hai góc tương ứng bằng nhau, theo định lý đồng dạng tam giác góc-góc (AA), ta có thể kết luận là \(\triangle ABD\) đồng dạng với \(\triangle CEB\). \( \therefore \triangle ABD \sim \triangle CEB \) (đpcm).
Parallel Lines and Transversals Angles
The image shows two parallel lines, labeled \( m \) and \( n \), with a transversal line crossing them. The angle formed between the transversal and line \( n \) is given as 133 degrees, and the angle formed between the transversal and line \( m \) is labeled as \( x \) degrees. Since lines \( m \) and \( n \) are parallel and are cut by a transversal, the corresponding angles are congruent. This means the angle opposite to the 133-degree angle (across from the transversal) will also be 133 degrees because it is a corresponding angle to the one on line \( n \). Therefore, the value of \( x \) is 133 degrees.
Parallel Lines and Corresponding Angles
The given diagram shows two parallel lines, marked as "l" and "m", and a transversal that intersects these parallel lines creating corresponding angles and alternate interior angles. In the diagram, there is a 140° angle formed between the transversal and line "m." Because line "l" is parallel to line "m," the angle formed between the transversal and line "l" that corresponds to the 140° angle will also be 140°. This is due to the Corresponding Angles Postulate, which states that when a transversal crosses parallel lines, corresponding angles are equal. Since we are given the notation m // n, where "m" and "n" are lines, we can assume "n" refers to the line labeled as "l" in the diagram, meaning that "l" and "m" are parallel. With this information, we can now determine that the value of angle x is also 140°, as it is a corresponding angle to the 140° angle on line "m". Therefore, x = 140°.
Solving for Angle BAC with Corresponding Angles
To solve for the measure of angle \( \angle BAC \) in the given image, you will need to use the properties of parallel lines and transversals. The image depicts two parallel lines cut by a transversal. The angles created by the transversal with the parallel lines are called corresponding angles when they occupy the same relative position. According to the Corresponding Angles Postulate, corresponding angles are congruent when the lines are parallel. If \( \angle B'AC' = 51° \), as stated in the image, then \( \angle BAC \), which is the corresponding angle to \( \angle B'AC' \), must also measure 51° because lines \( a \) and \( b \) are parallel, and line \( c \) serves as the transversal. Therefore, to the nearest tenth of a degree, the measure of angle \( \angle BAC \) is 51.0°.
Determining Parallel Lines Using Alternate Interior Angles
The image shows two lines cut by a transversal, with two angles labeled as \( m \angle 4 \) and \( m \angle 7 \). We are told that \( m \angle 4 = m \angle 7 \). This means angle 4 and angle 7 are congruent. Using this information, we can determine which lines are parallel. Congruent corresponding angles, which occur when a transversal crosses two parallel lines, imply that the lines are indeed parallel. In this case, angle 4 and angle 7 are alternate interior angles. According to the Alternate Interior Angles Theorem, if alternate interior angles are congruent, then the lines are parallel. So, the correct answer is: \[ \bar{t} || \bar{m}, \text{ by the Converse of the Alternate Interior Angles Theorem.} \] This is the third option in the list provided in the image.
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