what must the angles in a triange do to add up?
The angles in a triangle always add up to 180 degrees.
The angles in a triangle must always add up to 180 degrees.
The angles in a triangle always add up to a total of 180 degrees. This is a fundamental property of triangles.
To understand why the angles in a triangle add up to 180 degrees, we can use a geometric proof. Here's a step-by-step explanation:
1. Start with an arbitrary triangle ABC.
2. Draw a line, called a transversal, that intersects line segment BC at a point P, as shown:
A
/ \
/ \
/ \
B-------C
\
\
\
P
3. Extend line segments AB and AC to form two new lines running parallel to line segment BC, as shown:
A
/ \
/ \
B-------C
\
\
\
|\
| \
| \
D-------E---P
4. Since line segment AB is parallel to line segment CE and line segment AP intersects these two parallel lines, angle CAB is equal to angle CPE (alternate interior angles).
5. Similarly, since line segment AC is parallel to line segment BD and line segment AP intersects these two parallel lines, angle BAC is equal to angle BDP (alternate interior angles).
Now, let's calculate the sum of the angles in triangle ABC:
6. The three angles in triangle ABC are CAB, BAC, and CBA.
7. From steps 4 and 5, we know that angles CAB and BAC are equal to angles CPE and BDP, respectively.
8. Therefore, we have: CAB + BAC = CPE + BDP.
9. From the properties of parallel lines intersected by a transversal, we know that angle CPE and angle BDP form a straight line, so their sum is 180 degrees.
10. Therefore, CPE + BDP = 180 degrees.
11. Combining equations from steps 8 and 10, we have: CAB + BAC = 180 degrees.
Hence, the angles in a triangle always add up to 180 degrees. This relationship holds true for all triangles, regardless of their size or shape.