and
are perpendiculars dropped from
to
, and
is perpendicular to
and
. If CR = DS, which statement must be true?
The figure shows AB and PQ are parallel lines. CR and DS intersect the two lines. R and S have an angle of 90 degrees.
A.
m∠RCD = m∠SDB ÷ 2
B.
m∠RCD = m∠ACD
C.
m∠RCD = m∠ACD ÷ 2
D.
m∠RCD = m∠ACD ÷ 3
E.
m∠RCD = m∠ACD × 2
The correct answer is E. m∠RCD = m∠ACD × 2.
Since CR and DS are perpendiculars to AB and PQ respectively, triangle CDR is similar to triangle ASD (by angles). Since CR = DS, this means that CD = AS.
Also, since CR and DS are perpendiculars, angle RCD and angle SDA are both 90 degrees.
By the similarity of triangles CDR and ASD, we have:
CD/DR = AS/SD
Since CD = AS, this means that DR = SD.
Therefore, in triangle RCD and triangle SDA, we have:
m∠RCD = m∠SDA
Since m∠SDA = m∠ACD (since angle SDA and angle ACD are corresponding angles), this means that:
m∠RCD = m∠ACD.
Finally, since DR = SD, this means that triangle RCD = triangle SDA, so:
m∠RCD = m∠ACD × 2.
Therefore, answer choice E is correct.
To determine which statement must be true, let's analyze the given information.
In the figure, AB and PQ are parallel lines. CR and DS are perpendiculars dropped from C and D to AB and PQ, respectively. Additionally, RS is perpendicular to both AB and PQ at points R and S, respectively.
We are given that CR = DS. This implies that the two perpendiculars are congruent.
Now, let's consider the angles involved:
Angle RCD is formed by the intersecting lines CR and AB.
Angle SDB is formed by the intersecting lines DS and PQ.
Angle ACD is the angle formed at point C by AB and RS.
Based on the given information, we can conclude that Angle RCD is congruent to Angle SDB (since CR = DS).
Now, based on the properties of parallel lines, we know that alternate interior angles (like Angle ACD) are congruent when intersected by a transversal (like RS).
Therefore, we can conclude that Angle RCD is congruent to Angle ACD.
Applying what we found, we can deduce that statement B is true: m∠RCD = m∠ACD.
Hence, the correct answer is B.
To determine which statement is true, we can use the given information about the perpendiculars and the intersecting lines.
Given: CR is perpendicular to AB, DS is perpendicular to PQ, CR intersects DS, R and S have an angle of 90 degrees.
Using the properties of perpendicular lines, we know that opposite angles formed by the intersection of two perpendicular lines are congruent. In this case, angle RCD is congruent to angle SDB.
We are also given that CR = DS. This means that the lengths of the line segments CR and DS are equal.
Now, let's evaluate each statement:
A. m∠RCD = m∠SDB ÷ 2
B. m∠RCD = m∠ACD
C. m∠RCD = m∠ACD ÷ 2
D. m∠RCD = m∠ACD ÷ 3
E. m∠RCD = m∠ACD × 2
Statement A states that angle RCD is equal to half of angle SDB. We know that angle RCD and angle SDB are congruent, so this statement is true. However, it does not provide any relationship between angle RCD and angle ACD.
Statement B states that angle RCD is equal to angle ACD. This statement does not provide any relationship between angle RCD and angle ACD, so it is not necessarily true.
Statement C states that angle RCD is equal to half of angle ACD. Since angle RCD is congruent to angle SDB, and we know that angle SDB is congruent to angle ACD (since they are opposite angles formed by the intersection of perpendicular lines), this statement is true.
Statement D states that angle RCD is equal to one-third of angle ACD. Since we have no given information or reasoning to support this relationship, this statement is not necessarily true.
Statement E states that angle RCD is equal to twice angle ACD. Again, we have no given information or reasoning to support this relationship, so this statement is not necessarily true.
Based on our evaluation, the statement that must be true is:
C. m∠RCD = m∠ACD ÷ 2