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my book says
if a1 is perpendicular to a2 and b1 is perpeindicular to b2 then angle one equal angle two
WHY????
Also I remeber it was very simple but don't remeber why... if we exten lines a2 and b2 then angle two equals angle 3
WHY???
Please provide me with the reasons...
Thanks
I see two right-angled triangles in which opposite angles are also equal.
So if two angles of one triangle are equal to two angles of another triangle then the third angle pair must also be equal.
"sum of angles in a triangle theorem"
what are these opposite angles that you are making reference to sorry it's been a while sense I took geometry
draw any two intersecting straight lines.
Do you see two pairs of opposing or "opposite" angles?
Aren't they equal ?
Opposite angles are equal because each of those angles is supplementary to a common angle.
A+B=180
A+C=180
B must =C
oh and also what makes angle three equal to angle to if you extended the lines
Those will be opposite angles (if two lines intersect, then opposite angles are equal)
To understand why angle one equals angle two when a1 is perpendicular to a2 and b1 is perpendicular to b2, we need to consider the properties of perpendicular lines and angles.
First, let's establish some definitions:
- Perpendicular lines: Two lines are perpendicular if they intersect at a 90-degree angle.
- Perpendicular angles: When two lines intersect, the angles opposite each other (i.e., on opposite sides of the intersection) are called perpendicular angles.
Now, let's examine the given situation:
1. If a1 is perpendicular to a2:
- This means that line a1 intersects line a2 at a 90-degree angle. Let's call this intersection point P.
- Now, let's draw two more lines, b1 and b2, such that b1 is perpendicular to b2. Let's call their intersection point Q.
We have two cases to consider:
Case 1: Angle one (formed by a1 and b1) is equal to angle two (formed by a2 and b2).
- To prove this, we can use the property that if a line is perpendicular to two other lines, then the angles formed between these lines are congruent (i.e., equal in measure).
- So, angle one and angle two are congruent because both a1 and a2 are perpendicular to b1 and b2, respectively.
Case 2: Angle two equals angle three (formed by a2 and the extension of b2).
- To prove this, we need to extend line b2. Let's extend it to a point R.
- Since a1 and b1, as well as a2 and b2, are perpendicular to each other, they form congruent angles.
- Hence, angle two, formed by a2 and b2, is congruent to angle three, formed by a2 and the extension of b2.
In both cases, we have shown that the given angles are congruent (or equal) using the given information. These properties are derived from the definition of perpendicular lines and angles.
I hope this explanation helps clarify why angle one equals angle two and angle two equals angle three in the given scenario.